powComplex, real part

Percentage Accurate: 40.5% → 78.4%

Time: 16.9s

Alternatives: 17

Speedup: 5.9×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, t\_1 \cdot y.im\right)\right)\\ t_4 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_5 := t\_4 \cdot y.im\\ \mathbf{if}\;y.re \leq -41000:\\ \;\;\;\;\mathsf{fma}\left(-y.re, \sin t\_5 \cdot \tan^{-1}_* \frac{x.im}{x.re}, \cos t\_5\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 116000000:\\ \;\;\;\;\frac{\left(\cos \left(\mathsf{fma}\left(y.im, t\_1, t\_2\right)\right) \cdot t\_3\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin t\_2 \cdot t\_4, \cos t\_2\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re)))))
        (t_1 (log (hypot x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.re))
        (t_3 (cos (fma (- (atan2 x.im x.re)) y.re (* t_1 y.im))))
        (t_4 (log (hypot x.re x.im)))
        (t_5 (* t_4 y.im)))
   (if (<= y.re -41000.0)
     (* (fma (- y.re) (* (sin t_5) (atan2 x.im x.re)) (cos t_5)) t_0)
     (if (<= y.re 116000000.0)
       (/
        (* (* (cos (fma y.im t_1 t_2)) t_3) (pow (hypot x.im x.re) y.re))
        (* (pow (exp y.im) (atan2 x.im x.re)) t_3))
       (* (fma (- y.im) (* (sin t_2) t_4) (cos t_2)) 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_1 = log(hypot(x_46_im, x_46_re));
	double t_2 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_3 = cos(fma(-atan2(x_46_im, x_46_re), y_46_re, (t_1 * y_46_im)));
	double t_4 = log(hypot(x_46_re, x_46_im));
	double t_5 = t_4 * y_46_im;
	double tmp;
	if (y_46_re <= -41000.0) {
		tmp = fma(-y_46_re, (sin(t_5) * atan2(x_46_im, x_46_re)), cos(t_5)) * t_0;
	} else if (y_46_re <= 116000000.0) {
		tmp = ((cos(fma(y_46_im, t_1, t_2)) * t_3) * pow(hypot(x_46_im, x_46_re), y_46_re)) / (pow(exp(y_46_im), atan2(x_46_im, x_46_re)) * t_3);
	} else {
		tmp = fma(-y_46_im, (sin(t_2) * t_4), cos(t_2)) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_1 = log(hypot(x_46_im, x_46_re))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_3 = cos(fma(Float64(-atan(x_46_im, x_46_re)), y_46_re, Float64(t_1 * y_46_im)))
	t_4 = log(hypot(x_46_re, x_46_im))
	t_5 = Float64(t_4 * y_46_im)
	tmp = 0.0
	if (y_46_re <= -41000.0)
		tmp = Float64(fma(Float64(-y_46_re), Float64(sin(t_5) * atan(x_46_im, x_46_re)), cos(t_5)) * t_0);
	elseif (y_46_re <= 116000000.0)
		tmp = Float64(Float64(Float64(cos(fma(y_46_im, t_1, t_2)) * t_3) * (hypot(x_46_im, x_46_re) ^ y_46_re)) / Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) * t_3));
	else
		tmp = Float64(fma(Float64(-y_46_im), Float64(sin(t_2) * t_4), cos(t_2)) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$re + N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -41000.0], N[(N[((-y$46$re) * N[(N[Sin[t$95$5], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[Cos[t$95$5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 116000000.0], N[(N[(N[(N[Cos[N[(y$46$im * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[((-y$46$im) * N[(N[Sin[t$95$2], $MachinePrecision] * t$95$4), $MachinePrecision] + N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -41000

   1. Initial program 47.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if -41000 < y.re < 1.16e8

   1. Initial program 49.2%

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

   4. Applied rewrites 82.4%

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

   if 1.16e8 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -41000:\\ \;\;\;\;\mathsf{fma}\left(-y.re, \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 116000000:\\ \;\;\;\;\frac{\left(\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := t\_2 \cdot y.im\\ t_4 := \cos t\_3\\ \mathbf{if}\;y.re \leq -0.00175:\\ \;\;\;\;\mathsf{fma}\left(-y.re, \sin t\_3 \cdot \tan^{-1}_* \frac{x.im}{x.re}, t\_4\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 0.0003:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin t\_1 \cdot t\_2, \cos t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re)))))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (* t_2 y.im))
        (t_4 (cos t_3)))
   (if (<= y.re -0.00175)
     (* (fma (- y.re) (* (sin t_3) (atan2 x.im x.re)) t_4) t_0)
     (if (<= y.re 0.0003)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_4)
       (* (fma (- y.im) (* (sin t_1) t_2) (cos t_1)) 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = t_2 * y_46_im;
	double t_4 = cos(t_3);
	double tmp;
	if (y_46_re <= -0.00175) {
		tmp = fma(-y_46_re, (sin(t_3) * atan2(x_46_im, x_46_re)), t_4) * t_0;
	} else if (y_46_re <= 0.0003) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_4;
	} else {
		tmp = fma(-y_46_im, (sin(t_1) * t_2), cos(t_1)) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = Float64(t_2 * y_46_im)
	t_4 = cos(t_3)
	tmp = 0.0
	if (y_46_re <= -0.00175)
		tmp = Float64(fma(Float64(-y_46_re), Float64(sin(t_3) * atan(x_46_im, x_46_re)), t_4) * t_0);
	elseif (y_46_re <= 0.0003)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_4);
	else
		tmp = Float64(fma(Float64(-y_46_im), Float64(sin(t_1) * t_2), cos(t_1)) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * y$46$im), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, If[LessEqual[y$46$re, -0.00175], N[(N[((-y$46$re) * N[(N[Sin[t$95$3], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 0.0003], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[((-y$46$im) * N[(N[Sin[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision] + N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.00175000000000000004

   1. Initial program 46.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 86.8%

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

   if -0.00175000000000000004 < y.re < 2.99999999999999974e-4

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. prod-exp N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-atan2.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if 2.99999999999999974e-4 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00175:\\ \;\;\;\;\mathsf{fma}\left(-y.re, \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.0003:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \cos t\_1\\ t_3 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{elif}\;y.re \leq 0.0003:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(t\_0 \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin t\_1 \cdot t\_0, t\_2\right) \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im)))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (cos t_1))
        (t_3
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re))))))
   (if (<= y.re -1.25e-8)
     (* t_2 t_3)
     (if (<= y.re 0.0003)
       (* (exp (* (- y.im) (atan2 x.im x.re))) (cos (* t_0 y.im)))
       (* (fma (- y.im) (* (sin t_1) t_0) t_2) t_3)))))

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));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = cos(t_1);
	double t_3 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.25e-8) {
		tmp = t_2 * t_3;
	} else if (y_46_re <= 0.0003) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((t_0 * y_46_im));
	} else {
		tmp = fma(-y_46_im, (sin(t_1) * t_0), t_2) * t_3;
	}
	return tmp;
}

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))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = cos(t_1)
	t_3 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.25e-8)
		tmp = Float64(t_2 * t_3);
	elseif (y_46_re <= 0.0003)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(t_0 * y_46_im)));
	else
		tmp = Float64(fma(Float64(-y_46_im), Float64(sin(t_1) * t_0), t_2) * t_3);
	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[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e-8], N[(t$95$2 * t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 0.0003], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-y$46$im) * N[(N[Sin[t$95$1], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.2499999999999999e-8

   1. Initial program 46.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.2499999999999999e-8 < y.re < 2.99999999999999974e-4

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. prod-exp N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-atan2.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if 2.99999999999999974e-4 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.0003:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 0.0095:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re)))))
        (t_1 (cos (* (log (hypot x.re x.im)) y.im))))
   (if (<= y.re -1.25e-8)
     (* (cos (* (atan2 x.im x.re) y.re)) t_0)
     (if (<= y.re 0.0095)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_1)
       (* t_1 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_1 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	double tmp;
	if (y_46_re <= -1.25e-8) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	} else if (y_46_re <= 0.0095) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	} else {
		tmp = t_1 * 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double t_1 = Math.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	double tmp;
	if (y_46_re <= -1.25e-8) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	} else if (y_46_re <= 0.0095) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_1;
	} else {
		tmp = t_1 * 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	t_1 = math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	tmp = 0
	if y_46_re <= -1.25e-8:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * t_0
	elif y_46_re <= 0.0095:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_1
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_1 = cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.25e-8)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * t_0);
	elseif (y_46_re <= 0.0095)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_1);
	else
		tmp = Float64(t_1 * 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_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	t_1 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.25e-8)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	elseif (y_46_re <= 0.0095)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	else
		tmp = t_1 * 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[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e-8], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 0.0095], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.2499999999999999e-8

   1. Initial program 46.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.2499999999999999e-8 < y.re < 0.00949999999999999976

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. prod-exp N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-atan2.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if 0.00949999999999999976 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.0095:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.25e-8)
   (*
    (cos (* (atan2 x.im x.re) y.re))
    (exp
     (-
      (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
      (* y.im (atan2 x.im x.re)))))
   (if (<= y.re 9.5e-15)
     (*
      (exp (* (- y.im) (atan2 x.im x.re)))
      (cos (* (log (hypot x.re x.im)) y.im)))
     (* 1.0 (pow (hypot x.re x.im) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.25e-8) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 9.5e-15) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	} else {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.25e-8) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 9.5e-15) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	} else {
		tmp = 1.0 * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.25e-8:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	elif y_46_re <= 9.5e-15:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	else:
		tmp = 1.0 * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.25e-8)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (y_46_re <= 9.5e-15)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)));
	else
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.25e-8)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	elseif (y_46_re <= 9.5e-15)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	else
		tmp = 1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.25e-8], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-15], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.2499999999999999e-8

   1. Initial program 46.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.2499999999999999e-8 < y.re < 9.5000000000000005e-15

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x.re around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. prod-exp N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-atan2.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if 9.5000000000000005e-15 < y.re

   1. Initial program 37.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.9%

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -0.00037:\\ \;\;\;\;t\_0 \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re)))
   (if (<= y.re -0.00037)
     (* t_0 (cos (* (atan2 x.im x.re) y.re)))
     (if (<= y.re 9.5e-15)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (* (log (hypot x.re x.im)) y.im)))
       (* 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 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -0.00037) {
		tmp = t_0 * cos((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= 9.5e-15) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	} else {
		tmp = 1.0 * 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.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -0.00037) {
		tmp = t_0 * Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= 9.5e-15) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -0.00037:
		tmp = t_0 * math.cos((math.atan2(x_46_im, x_46_re) * y_46_re))
	elif y_46_re <= 9.5e-15:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	else:
		tmp = 1.0 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -0.00037)
		tmp = Float64(t_0 * cos(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_re <= 9.5e-15)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)));
	else
		tmp = Float64(1.0 * 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 = hypot(x_46_re, x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -0.00037)
		tmp = t_0 * cos((atan2(x_46_im, x_46_re) * y_46_re));
	elseif (y_46_re <= 9.5e-15)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	else
		tmp = 1.0 * 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[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -0.00037], N[(t$95$0 * N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-15], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.6999999999999999e-4

   1. Initial program 46.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -3.6999999999999999e-4 < y.re < 9.5000000000000005e-15

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x.re around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. prod-exp N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-atan2.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if 9.5000000000000005e-15 < y.re

   1. Initial program 37.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.9%

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00037:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (cos (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -1.05e-8)
     (* t_0 t_1)
     (if (<= y.re 9.5e-15)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_1)
       (* 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 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = cos((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -1.05e-8) {
		tmp = t_0 * t_1;
	} else if (y_46_re <= 9.5e-15) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	} else {
		tmp = 1.0 * 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.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -1.05e-8) {
		tmp = t_0 * t_1;
	} else if (y_46_re <= 9.5e-15) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_1;
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_1 = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -1.05e-8:
		tmp = t_0 * t_1
	elif y_46_re <= 9.5e-15:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_1
	else:
		tmp = 1.0 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = cos(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.05e-8)
		tmp = Float64(t_0 * t_1);
	elseif (y_46_re <= 9.5e-15)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_1);
	else
		tmp = Float64(1.0 * 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 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_1 = cos((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -1.05e-8)
		tmp = t_0 * t_1;
	elseif (y_46_re <= 9.5e-15)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	else
		tmp = 1.0 * 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[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e-8], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-15], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.04999999999999997e-8

   1. Initial program 46.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -1.04999999999999997e-8 < y.re < 9.5000000000000005e-15

   1. Initial program 50.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in y.im around inf

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 80.7

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

   8. Applied rewrites 80.7%

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

   if 9.5000000000000005e-15 < y.re

   1. Initial program 37.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.9%

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-8}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.6e+18)
   (*
    (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) x.re) y.re)
    (cos (* (atan2 x.im x.re) y.re)))
   (* 1.0 (pow (hypot x.re x.im) y.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_im <= -4.6e+18) {
		tmp = pow((fma((0.5 / x_46_re), ((x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re), y_46_re) * cos((atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_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_im <= -4.6e+18)
		tmp = Float64((Float64(fma(Float64(0.5 / x_46_re), Float64(Float64(x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re) ^ y_46_re) * cos(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.6e+18], N[(N[Power[N[(N[(N[(0.5 / x$46$re), $MachinePrecision] * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + 1.0), $MachinePrecision] * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.6e18

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 36.7%

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

   if -4.6e18 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.6e+18)
   (*
    (pow (fma (/ (* x.im x.im) x.re) 0.5 x.re) y.re)
    (cos (* (atan2 x.im x.re) y.re)))
   (* 1.0 (pow (hypot x.re x.im) y.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_im <= -4.6e+18) {
		tmp = pow(fma(((x_46_im * x_46_im) / x_46_re), 0.5, x_46_re), y_46_re) * cos((atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_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_im <= -4.6e+18)
		tmp = Float64((fma(Float64(Float64(x_46_im * x_46_im) / x_46_re), 0.5, x_46_re) ^ y_46_re) * cos(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.6e+18], N[(N[Power[N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] * 0.5 + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.6e18

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 35.3%

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

   if -4.6e18 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -0.096:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 0.0009:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re) 1.0)))
   (if (<= y.re -0.096)
     t_0
     (if (<= y.re 0.0009) (fma (log (hypot x.re x.im)) y.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 = pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -0.096) {
		tmp = t_0;
	} else if (y_46_re <= 0.0009) {
		tmp = fma(log(hypot(x_46_re, x_46_im)), y_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((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -0.096)
		tmp = t_0;
	elseif (y_46_re <= 0.0009)
		tmp = fma(log(hypot(x_46_re, x_46_im)), y_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[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -0.096], t$95$0, If[LessEqual[y$46$re, 0.0009], N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.096000000000000002 or 8.9999999999999998e-4 < y.re

   1. Initial program 42.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.5%

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

   12. Taylor expanded in x.im around 0

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

   14. Applied rewrites 69.2%

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

   if -0.096000000000000002 < y.re < 8.9999999999999998e-4

   1. Initial program 49.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 47.7%

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

Alternative 11: 64.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+20}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.12e+20)
   (* (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) x.re) y.re) 1.0)
   (* 1.0 (pow (hypot x.re x.im) y.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_im <= -1.12e+20) {
		tmp = pow((fma((0.5 / x_46_re), ((x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re), y_46_re) * 1.0;
	} else {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_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_im <= -1.12e+20)
		tmp = Float64((Float64(fma(Float64(0.5 / x_46_re), Float64(Float64(x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re) ^ y_46_re) * 1.0);
	else
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.12e+20], N[(N[Power[N[(N[(N[(0.5 / x$46$re), $MachinePrecision] * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + 1.0), $MachinePrecision] * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.12e20

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 26.0%

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

   12. Taylor expanded in x.re around inf

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

   14. Applied rewrites 32.3%

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

   if -1.12e20 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+20}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -0.096:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.08 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re) 1.0)))
   (if (<= y.re -0.096) t_0 (if (<= y.re 1.08e-32) 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 = pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -0.096) {
		tmp = t_0;
	} else if (y_46_re <= 1.08e-32) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -0.096)
		tmp = t_0;
	elseif (y_46_re <= 1.08e-32)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -0.096], t$95$0, If[LessEqual[y$46$re, 1.08e-32], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.096000000000000002 or 1.08e-32 < y.re

   1. Initial program 42.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 66.2%

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

   12. Taylor expanded in x.im around 0

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

   14. Applied rewrites 68.6%

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

   if -0.096000000000000002 < y.re < 1.08e-32

   1. Initial program 50.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 47.6%

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

Alternative 13: 55.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.14 \cdot 10^{-74}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 9.2 \cdot 10^{-72}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.14e-74)
   (* (pow (- x.re) y.re) 1.0)
   (if (<= x.re 9.2e-72) (* (pow (- x.im) y.re) 1.0) (* (pow x.re y.re) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.14e-74) {
		tmp = pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 9.2e-72) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-1.14d-74)) then
        tmp = (-x_46re ** y_46re) * 1.0d0
    else if (x_46re <= 9.2d-72) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else
        tmp = (x_46re ** y_46re) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.14e-74) {
		tmp = Math.pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 9.2e-72) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -1.14e-74:
		tmp = math.pow(-x_46_re, y_46_re) * 1.0
	elif x_46_re <= 9.2e-72:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.14e-74)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * 1.0);
	elseif (x_46_re <= 9.2e-72)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.14e-74)
		tmp = (-x_46_re ^ y_46_re) * 1.0;
	elseif (x_46_re <= 9.2e-72)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	else
		tmp = (x_46_re ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1.14e-74], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 9.2e-72], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.13999999999999993e-74

   1. Initial program 35.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 32.5%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 38.2%

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

   12. Taylor expanded in x.re around -inf

       \[\leadsto {\left(-1 \cdot x.re\right)}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 56.1%

       \[\leadsto {\left(-x.re\right)}^{y.re} \cdot 1 \]

   if -1.13999999999999993e-74 < x.re < 9.19999999999999978e-72

   1. Initial program 57.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 43.8%

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

   12. Taylor expanded in x.im around -inf

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

   14. Applied rewrites 55.7%

       \[\leadsto {\left(-x.im\right)}^{y.re} \cdot 1 \]

   if 9.19999999999999978e-72 < x.re

   1. Initial program 41.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 39.4%

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

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 60.5%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.85:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.85)
   (* (pow x.im y.re) 1.0)
   (if (<= y.re 1.8e-7) 1.0 (* (pow x.re y.re) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.85) {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	} else if (y_46_re <= 1.8e-7) {
		tmp = 1.0;
	} else {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.85d0)) then
        tmp = (x_46im ** y_46re) * 1.0d0
    else if (y_46re <= 1.8d-7) then
        tmp = 1.0d0
    else
        tmp = (x_46re ** y_46re) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.85) {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	} else if (y_46_re <= 1.8e-7) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.85:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	elif y_46_re <= 1.8e-7:
		tmp = 1.0
	else:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.85)
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	elseif (y_46_re <= 1.8e-7)
		tmp = 1.0;
	else
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.85)
		tmp = (x_46_im ^ y_46_re) * 1.0;
	elseif (y_46_re <= 1.8e-7)
		tmp = 1.0;
	else
		tmp = (x_46_re ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.85], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-7], 1.0, N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.8500000000000001

   1. Initial program 46.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 72.6%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

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

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 60.4%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]

   if -1.8500000000000001 < y.re < 1.79999999999999997e-7

   1. Initial program 49.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 47.6%

      \[\leadsto 1 \]

   if 1.79999999999999997e-7 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 54.7%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 63.5%

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

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 61.8%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -1.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.im y.re) 1.0)))
   (if (<= y.re -1.85) t_0 (if (<= y.re 2.8) 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 = pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -1.85) {
		tmp = t_0;
	} else if (y_46_re <= 2.8) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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 = (x_46im ** y_46re) * 1.0d0
    if (y_46re <= (-1.85d0)) then
        tmp = t_0
    else if (y_46re <= 2.8d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -1.85) {
		tmp = t_0;
	} else if (y_46_re <= 2.8) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_im, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -1.85:
		tmp = t_0
	elif y_46_re <= 2.8:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_im ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -1.85)
		tmp = t_0;
	elseif (y_46_re <= 2.8)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -1.85)
		tmp = t_0;
	elseif (y_46_re <= 2.8)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -1.85], t$95$0, If[LessEqual[y$46$re, 2.8], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.8500000000000001 or 2.7999999999999998 < y.re

   1. Initial program 42.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.5%

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

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 60.2%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]

   if -1.8500000000000001 < y.re < 2.7999999999999998

   1. Initial program 49.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 47.6%

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

Alternative 16: 53.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{-307}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -8e-307) (* (pow (- x.im) y.re) 1.0) (* (pow x.im y.re) 1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -8e-307) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-8d-307)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -8e-307) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -8e-307:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -8e-307)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -8e-307)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -8e-307], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -7.99999999999999927e-307

   1. Initial program 47.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 33.2%

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

   12. Taylor expanded in x.im around -inf

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

   14. Applied rewrites 57.7%

       \[\leadsto {\left(-x.im\right)}^{y.re} \cdot 1 \]

   if -7.99999999999999927e-307 < x.im

   1. Initial program 44.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{x.im}, \frac{x.re \cdot x.re}{x.im}, 1\right) \cdot x.im\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 49.8%

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

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 49.8%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.6% accurate, 680.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 46.4%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-atan2.f64 59.8

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

5. Applied rewrites 59.8%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 26.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))