powComplex, real part

Percentage Accurate: 41.7% → 77.8%

Time: 18.7s

Alternatives: 17

Speedup: 5.7×

• 35.6% 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: 41.7% 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: 77.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\\ t_2 := \log t\_1 \cdot y.im\\ t_3 := \cos t\_0 \cdot \cos t\_2\\ t_4 := \sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\\ t_5 := \sin t\_2 \cdot \sin t\_0\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\log t\_4 \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left({t\_3}^{3} - {t\_5}^{3}\right) \cdot {t\_1}^{y.re}}{\mathsf{fma}\left(t\_5, \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, t\_2\right)\right), {t\_3}^{2}\right) \cdot {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {t\_4}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1 (sqrt (hypot x.im x.re)))
        (t_2 (* (log t_1) y.im))
        (t_3 (* (cos t_0) (cos t_2)))
        (t_4 (sqrt (hypot x.re x.im)))
        (t_5 (* (sin t_2) (sin t_0))))
   (if (<= y.re -2e-9)
     (*
      (cos (* (log t_4) y.im))
      (exp
       (-
        (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
        (* y.im (atan2 x.im x.re)))))
     (if (<= y.re 3.9e+41)
       (/
        (* (- (pow t_3 3.0) (pow t_5 3.0)) (pow t_1 y.re))
        (*
         (fma t_5 (cos (fma (- (atan2 x.im x.re)) y.re t_2)) (pow t_3 2.0))
         (pow (exp y.im) (atan2 x.im x.re))))
       (* 1.0 (pow t_4 y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sqrt(hypot(x_46_im, x_46_re));
	double t_2 = log(t_1) * y_46_im;
	double t_3 = cos(t_0) * cos(t_2);
	double t_4 = sqrt(hypot(x_46_re, x_46_im));
	double t_5 = sin(t_2) * sin(t_0);
	double tmp;
	if (y_46_re <= -2e-9) {
		tmp = cos((log(t_4) * y_46_im)) * 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 <= 3.9e+41) {
		tmp = ((pow(t_3, 3.0) - pow(t_5, 3.0)) * pow(t_1, y_46_re)) / (fma(t_5, cos(fma(-atan2(x_46_im, x_46_re), y_46_re, t_2)), pow(t_3, 2.0)) * pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * pow(t_4, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sqrt(hypot(x_46_im, x_46_re))
	t_2 = Float64(log(t_1) * y_46_im)
	t_3 = Float64(cos(t_0) * cos(t_2))
	t_4 = sqrt(hypot(x_46_re, x_46_im))
	t_5 = Float64(sin(t_2) * sin(t_0))
	tmp = 0.0
	if (y_46_re <= -2e-9)
		tmp = Float64(cos(Float64(log(t_4) * y_46_im)) * 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 <= 3.9e+41)
		tmp = Float64(Float64(Float64((t_3 ^ 3.0) - (t_5 ^ 3.0)) * (t_1 ^ y_46_re)) / Float64(fma(t_5, cos(fma(Float64(-atan(x_46_im, x_46_re)), y_46_re, t_2)), (t_3 ^ 2.0)) * (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * (t_4 ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2e-9], N[(N[Cos[N[(N[Log[t$95$4], $MachinePrecision] * y$46$im), $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, 3.9e+41], N[(N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] - N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, y$46$re], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$5 * N[Cos[N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$re + t$95$2), $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[t$95$4, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.00000000000000012e-9

   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 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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -2.00000000000000012e-9 < y.re < 3.8999999999999997e41

   1. Initial program 40.7%

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

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re} \cdot \left({\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right)}^{3} - {\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right)}^{3}\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \mathsf{fma}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right), \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right), {\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right)}^{2}\right)}} \]

   if 3.8999999999999997e41 < y.re

   1. Initial program 34.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 82.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left({\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\right)}^{3} - {\left(\sin \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{3}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re}}{\mathsf{fma}\left(\sin \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right), \cos \left(\mathsf{fma}\left(-\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\right), {\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\right)}^{2}\right) \cdot {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{if}\;\cos \left(y.im \cdot t\_0 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{t\_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq 0:\\ \;\;\;\;{\left(\sqrt{\frac{{x.re}^{4} - {x.im}^{4}}{\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)}}\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.im x.im) (* x.re x.re))))))
   (if (<=
        (*
         (cos (+ (* y.im t_0) (* (atan2 x.im x.re) y.re)))
         (exp (- (* t_0 y.re) (* y.im (atan2 x.im x.re)))))
        0.0)
     (*
      (pow
       (sqrt
        (/ (- (pow x.re 4.0) (pow x.im 4.0)) (* (- x.re x.im) (+ x.im x.re))))
       y.re)
      1.0)
     (* 1.0 (pow (sqrt (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 t_0 = log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if ((cos(((y_46_im * t_0) + (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((t_0 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))))) <= 0.0) {
		tmp = pow(sqrt(((pow(x_46_re, 4.0) - pow(x_46_im, 4.0)) / ((x_46_re - x_46_im) * (x_46_im + x_46_re)))), y_46_re) * 1.0;
	} else {
		tmp = 1.0 * pow(sqrt(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 t_0 = Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if ((Math.cos(((y_46_im * t_0) + (Math.atan2(x_46_im, x_46_re) * y_46_re))) * Math.exp(((t_0 * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))))) <= 0.0) {
		tmp = Math.pow(Math.sqrt(((Math.pow(x_46_re, 4.0) - Math.pow(x_46_im, 4.0)) / ((x_46_re - x_46_im) * (x_46_im + x_46_re)))), y_46_re) * 1.0;
	} else {
		tmp = 1.0 * Math.pow(Math.sqrt(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):
	t_0 = math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	tmp = 0
	if (math.cos(((y_46_im * t_0) + (math.atan2(x_46_im, x_46_re) * y_46_re))) * math.exp(((t_0 * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))) <= 0.0:
		tmp = math.pow(math.sqrt(((math.pow(x_46_re, 4.0) - math.pow(x_46_im, 4.0)) / ((x_46_re - x_46_im) * (x_46_im + x_46_re)))), y_46_re) * 1.0
	else:
		tmp = 1.0 * math.pow(math.sqrt(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)
	t_0 = log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(y_46_im * t_0) + Float64(atan(x_46_im, x_46_re) * y_46_re))) * exp(Float64(Float64(t_0 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))) <= 0.0)
		tmp = Float64((sqrt(Float64(Float64((x_46_re ^ 4.0) - (x_46_im ^ 4.0)) / Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)))) ^ y_46_re) * 1.0);
	else
		tmp = Float64(1.0 * (sqrt(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)
	t_0 = log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	tmp = 0.0;
	if ((cos(((y_46_im * t_0) + (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((t_0 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))))) <= 0.0)
		tmp = (sqrt((((x_46_re ^ 4.0) - (x_46_im ^ 4.0)) / ((x_46_re - x_46_im) * (x_46_im + x_46_re)))) ^ y_46_re) * 1.0;
	else
		tmp = 1.0 * (sqrt(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_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(y$46$im * t$95$0), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[N[Sqrt[N[(N[(N[Power[x$46$re, 4.0], $MachinePrecision] - N[Power[x$46$im, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 0.0

   1. Initial program 79.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 59.0%

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

   10. Applied rewrites 67.6%

       \[\leadsto {\left(\sqrt{\frac{{x.re}^{4} - {x.im}^{4}}{\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)}}\right)}^{y.re} \cdot 1 \]

   if 0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 22.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 63.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq 0:\\ \;\;\;\;{\left(\sqrt{\frac{{x.re}^{4} - {x.im}^{4}}{\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)}}\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\log t\_0 \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {t\_0}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (hypot x.re x.im))))
   (if (<= y.re -2e-9)
     (*
      (cos (* (log t_0) y.im))
      (exp
       (-
        (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
        (* y.im (atan2 x.im x.re)))))
     (if (<= y.re 4.3e-23)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (* (log (sqrt (hypot x.im x.re))) y.im)))
       (* 1.0 (pow t_0 y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sqrt(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -2e-9) {
		tmp = cos((log(t_0) * y_46_im)) * 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 <= 4.3e-23) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = 1.0 * pow(t_0, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sqrt(Math.hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -2e-9) {
		tmp = Math.cos((Math.log(t_0) * y_46_im)) * 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 <= 4.3e-23) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((Math.log(Math.sqrt(Math.hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = 1.0 * Math.pow(t_0, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sqrt(math.hypot(x_46_re, x_46_im))
	tmp = 0
	if y_46_re <= -2e-9:
		tmp = math.cos((math.log(t_0) * y_46_im)) * 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 <= 4.3e-23:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((math.log(math.sqrt(math.hypot(x_46_im, x_46_re))) * y_46_im))
	else:
		tmp = 1.0 * math.pow(t_0, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= -2e-9)
		tmp = Float64(cos(Float64(log(t_0) * y_46_im)) * 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 <= 4.3e-23)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im)));
	else
		tmp = Float64(1.0 * (t_0 ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(hypot(x_46_re, x_46_im));
	tmp = 0.0;
	if (y_46_re <= -2e-9)
		tmp = cos((log(t_0) * y_46_im)) * 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 <= 4.3e-23)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	else
		tmp = 1.0 * (t_0 ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2e-9], N[(N[Cos[N[(N[Log[t$95$0], $MachinePrecision] * y$46$im), $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, 4.3e-23], 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[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[t$95$0, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.00000000000000012e-9

   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 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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -2.00000000000000012e-9 < y.re < 4.30000000000000002e-23

   1. Initial program 41.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-exp.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-atan2.f64 81.5

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

   8. Applied rewrites 81.5%

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

   if 4.30000000000000002e-23 < y.re

   1. Initial program 33.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 79.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re))))
   (if (<= y.re -4.9e+132)
     t_0
     (if (<= y.re -1.95e-9)
       (*
        (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 4.3e-23)
         (*
          (exp (* (- y.im) (atan2 x.im x.re)))
          (cos (* (log (sqrt (hypot x.im x.re))) y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -4.9e+132) {
		tmp = t_0;
	} else if (y_46_re <= -1.95e-9) {
		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 <= 4.3e-23) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -4.9e+132) {
		tmp = t_0;
	} else if (y_46_re <= -1.95e-9) {
		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 <= 4.3e-23) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((Math.log(Math.sqrt(Math.hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	tmp = 0
	if y_46_re <= -4.9e+132:
		tmp = t_0
	elif y_46_re <= -1.95e-9:
		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 <= 4.3e-23:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((math.log(math.sqrt(math.hypot(x_46_im, x_46_re))) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.9e+132)
		tmp = t_0;
	elseif (y_46_re <= -1.95e-9)
		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 <= 4.3e-23)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.9e+132)
		tmp = t_0;
	elseif (y_46_re <= -1.95e-9)
		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 <= 4.3e-23)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.9e+132], t$95$0, If[LessEqual[y$46$re, -1.95e-9], 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, 4.3e-23], 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[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.9000000000000002e132 or 4.30000000000000002e-23 < y.re

   1. Initial program 34.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 80.1%

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

   if -4.9000000000000002e132 < y.re < -1.9500000000000001e-9

   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

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -1.9500000000000001e-9 < y.re < 4.30000000000000002e-23

   1. Initial program 41.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-exp.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-atan2.f64 81.5

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

   8. Applied rewrites 81.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+132}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re))))
   (if (<= y.re -2.5)
     t_0
     (if (<= y.re 4.3e-23)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (* (log (sqrt (hypot x.im x.re))) y.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -2.5) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e-23) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -2.5) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e-23) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((Math.log(Math.sqrt(Math.hypot(x_46_im, x_46_re))) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	tmp = 0
	if y_46_re <= -2.5:
		tmp = t_0
	elif y_46_re <= 4.3e-23:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((math.log(math.sqrt(math.hypot(x_46_im, x_46_re))) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.5)
		tmp = t_0;
	elseif (y_46_re <= 4.3e-23)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.5)
		tmp = t_0;
	elseif (y_46_re <= 4.3e-23)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((log(sqrt(hypot(x_46_im, x_46_re))) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5], t$95$0, If[LessEqual[y$46$re, 4.3e-23], 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[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.5 or 4.30000000000000002e-23 < y.re

   1. Initial program 36.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 76.9%

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

   if -2.5 < y.re < 4.30000000000000002e-23

   1. Initial program 43.1%

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

   3. Taylor expanded in y.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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-exp.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-atan2.f64 81.0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re))))
   (if (<= y.re -2.7)
     t_0
     (if (<= y.re 4.3e-23)
       (*
        (cos (* (atan2 x.im x.re) y.re))
        (exp (* (- y.im) (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -2.7) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e-23) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -2.7) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e-23) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	tmp = 0
	if y_46_re <= -2.7:
		tmp = t_0
	elif y_46_re <= 4.3e-23:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.7)
		tmp = t_0;
	elseif (y_46_re <= 4.3e-23)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.7)
		tmp = t_0;
	elseif (y_46_re <= 4.3e-23)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.7], t$95$0, If[LessEqual[y$46$re, 4.3e-23], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.7000000000000002 or 4.30000000000000002e-23 < y.re

   1. Initial program 36.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 76.9%

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

   if -2.7000000000000002 < y.re < 4.30000000000000002e-23

   1. Initial program 43.1%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. exp-diff N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-atan2.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-atan2.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in y.im around 0

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

   11. Applied rewrites 78.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6 \cdot 10^{+210}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 6e+210)
   (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re))
   (* (pow x.im y.re) (cos (* (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 tmp;
	if (x_46_im <= 6e+210) {
		tmp = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	} else {
		tmp = pow(x_46_im, y_46_re) * cos((atan2(x_46_im, x_46_re) * 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 (x_46_im <= 6e+210) {
		tmp = 1.0 * Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= 6e+210:
		tmp = 1.0 * math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	else:
		tmp = math.pow(x_46_im, y_46_re) * math.cos((math.atan2(x_46_im, x_46_re) * y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= 6e+210)
		tmp = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re));
	else
		tmp = Float64((x_46_im ^ y_46_re) * cos(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= 6e+210)
		tmp = 1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re);
	else
		tmp = (x_46_im ^ y_46_re) * cos((atan2(x_46_im, x_46_re) * y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 6e+210], N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.00000000000000044e210

   1. Initial program 43.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 63.0%

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

   if 6.00000000000000044e210 < x.im

   1. Initial program 0.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6 \cdot 10^{+210}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-63}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.re \cdot x.re}{x.im}, 0.5, x.im\right)\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 0.0057:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right), y.re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e-63)
   (* (pow (fma (/ (* x.re x.re) x.im) 0.5 x.im) y.re) 1.0)
   (if (<= y.re 0.0057)
     (fma (log (sqrt (hypot x.re x.im))) y.re 1.0)
     (*
      (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) 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 <= -1.15e-63) {
		tmp = pow(fma(((x_46_re * x_46_re) / x_46_im), 0.5, x_46_im), y_46_re) * 1.0;
	} else if (y_46_re <= 0.0057) {
		tmp = fma(log(sqrt(hypot(x_46_re, x_46_im))), y_46_re, 1.0);
	} else {
		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;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e-63)
		tmp = Float64((fma(Float64(Float64(x_46_re * x_46_re) / x_46_im), 0.5, x_46_im) ^ y_46_re) * 1.0);
	elseif (y_46_re <= 0.0057)
		tmp = fma(log(sqrt(hypot(x_46_re, x_46_im))), y_46_re, 1.0);
	else
		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);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e-63], N[(N[Power[N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] * 0.5 + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 0.0057], N[(N[Log[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re + 1.0), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15e-63

   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 65.1%

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

   if -1.15e-63 < y.re < 0.0057000000000000002

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 46.8%

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

   if 0.0057000000000000002 < y.re

   1. Initial program 33.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 73.5%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -36000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -36000000.0)
   (* (pow (fma (/ (* x.im x.im) x.re) 0.5 x.re) y.re) 1.0)
   (* 1.0 (pow (sqrt (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 <= -36000000.0) {
		tmp = pow(fma(((x_46_im * x_46_im) / x_46_re), 0.5, x_46_re), y_46_re) * 1.0;
	} else {
		tmp = 1.0 * pow(sqrt(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 <= -36000000.0)
		tmp = Float64((fma(Float64(Float64(x_46_im * x_46_im) / x_46_re), 0.5, x_46_re) ^ y_46_re) * 1.0);
	else
		tmp = Float64(1.0 * (sqrt(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, -36000000.0], 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] * 1.0), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.6e7

   1. Initial program 38.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 41.0%

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

   if -3.6e7 < y.im

   1. Initial program 39.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 70.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -36000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-63}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.re \cdot x.re}{x.im}, 0.5, x.im\right)\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 0.0057:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e-63)
   (* (pow (fma (/ (* x.re x.re) x.im) 0.5 x.im) y.re) 1.0)
   (if (<= y.re 0.0057)
     1.0
     (*
      (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) 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 <= -1.15e-63) {
		tmp = pow(fma(((x_46_re * x_46_re) / x_46_im), 0.5, x_46_im), y_46_re) * 1.0;
	} else if (y_46_re <= 0.0057) {
		tmp = 1.0;
	} else {
		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;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e-63)
		tmp = Float64((fma(Float64(Float64(x_46_re * x_46_re) / x_46_im), 0.5, x_46_im) ^ y_46_re) * 1.0);
	elseif (y_46_re <= 0.0057)
		tmp = 1.0;
	else
		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);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e-63], N[(N[Power[N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] * 0.5 + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 0.0057], 1.0, 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]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15e-63

   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 65.1%

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

   if -1.15e-63 < y.re < 0.0057000000000000002

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto 1 \]

   if 0.0057000000000000002 < y.re

   1. Initial program 33.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 73.5%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-63}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.re \cdot x.re}{x.im}, 0.5, x.im\right)\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 0.0057:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e-63)
   (* (pow (fma (/ (* x.re x.re) x.im) 0.5 x.im) y.re) 1.0)
   (if (<= y.re 0.0057)
     1.0
     (* (pow (fma (/ (* x.im x.im) x.re) 0.5 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 <= -1.15e-63) {
		tmp = pow(fma(((x_46_re * x_46_re) / x_46_im), 0.5, x_46_im), y_46_re) * 1.0;
	} else if (y_46_re <= 0.0057) {
		tmp = 1.0;
	} else {
		tmp = pow(fma(((x_46_im * x_46_im) / x_46_re), 0.5, 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 <= -1.15e-63)
		tmp = Float64((fma(Float64(Float64(x_46_re * x_46_re) / x_46_im), 0.5, x_46_im) ^ y_46_re) * 1.0);
	elseif (y_46_re <= 0.0057)
		tmp = 1.0;
	else
		tmp = Float64((fma(Float64(Float64(x_46_im * x_46_im) / x_46_re), 0.5, x_46_re) ^ y_46_re) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e-63], N[(N[Power[N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] * 0.5 + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 0.0057], 1.0, 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] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15e-63

   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 65.1%

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

   if -1.15e-63 < y.re < 0.0057000000000000002

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto 1 \]

   if 0.0057000000000000002 < y.re

   1. Initial program 33.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 73.5%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re}\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 6 \cdot 10^{-189}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.35e-17)
   (* (pow (sqrt (* x.re x.re)) y.re) 1.0)
   (if (<= x.re 6e-189)
     (* (pow (* x.im x.im) (* 0.5 y.re)) 1.0)
     (* (pow (fma (/ (* x.im x.im) x.re) 0.5 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 <= -1.35e-17) {
		tmp = pow(sqrt((x_46_re * x_46_re)), y_46_re) * 1.0;
	} else if (x_46_re <= 6e-189) {
		tmp = pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0;
	} else {
		tmp = pow(fma(((x_46_im * x_46_im) / x_46_re), 0.5, 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 <= -1.35e-17)
		tmp = Float64((sqrt(Float64(x_46_re * x_46_re)) ^ y_46_re) * 1.0);
	elseif (x_46_re <= 6e-189)
		tmp = Float64((Float64(x_46_im * x_46_im) ^ Float64(0.5 * y_46_re)) * 1.0);
	else
		tmp = Float64((fma(Float64(Float64(x_46_im * x_46_im) / x_46_re), 0.5, x_46_re) ^ y_46_re) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1.35e-17], N[(N[Power[N[Sqrt[N[(x$46$re * x$46$re), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 6e-189], N[(N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], 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] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.3500000000000001e-17

   1. Initial program 32.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.6%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 55.7%

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

   if -1.3500000000000001e-17 < x.re < 6e-189

   1. Initial program 44.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.1%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 52.6%

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

   13. Applied rewrites 59.2%

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

   if 6e-189 < x.re

   1. Initial program 39.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 60.5%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re}\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 6 \cdot 10^{-189}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re}\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.35e-17)
   (* (pow (sqrt (* x.re x.re)) y.re) 1.0)
   (if (<= x.re 1.1e-26)
     (* (pow (* x.im x.im) (* 0.5 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 <= -1.35e-17) {
		tmp = pow(sqrt((x_46_re * x_46_re)), y_46_re) * 1.0;
	} else if (x_46_re <= 1.1e-26) {
		tmp = pow((x_46_im * x_46_im), (0.5 * 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 <= (-1.35d-17)) then
        tmp = (sqrt((x_46re * x_46re)) ** y_46re) * 1.0d0
    else if (x_46re <= 1.1d-26) then
        tmp = ((x_46im * x_46im) ** (0.5d0 * 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 <= -1.35e-17) {
		tmp = Math.pow(Math.sqrt((x_46_re * x_46_re)), y_46_re) * 1.0;
	} else if (x_46_re <= 1.1e-26) {
		tmp = Math.pow((x_46_im * x_46_im), (0.5 * 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 <= -1.35e-17:
		tmp = math.pow(math.sqrt((x_46_re * x_46_re)), y_46_re) * 1.0
	elif x_46_re <= 1.1e-26:
		tmp = math.pow((x_46_im * x_46_im), (0.5 * 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 <= -1.35e-17)
		tmp = Float64((sqrt(Float64(x_46_re * x_46_re)) ^ y_46_re) * 1.0);
	elseif (x_46_re <= 1.1e-26)
		tmp = Float64((Float64(x_46_im * x_46_im) ^ Float64(0.5 * 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 <= -1.35e-17)
		tmp = (sqrt((x_46_re * x_46_re)) ^ y_46_re) * 1.0;
	elseif (x_46_re <= 1.1e-26)
		tmp = ((x_46_im * x_46_im) ^ (0.5 * 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, -1.35e-17], N[(N[Power[N[Sqrt[N[(x$46$re * x$46$re), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 1.1e-26], N[(N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $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 < -1.3500000000000001e-17

   1. Initial program 32.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.6%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 55.7%

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

   if -1.3500000000000001e-17 < x.re < 1.1e-26

   1. Initial program 45.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 52.6%

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

   13. Applied rewrites 58.2%

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

   if 1.1e-26 < x.re

   1. Initial program 35.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 62.7%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re}\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-9}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 0.044:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -9.5e+101)
     t_0
     (if (<= y.re -2.25e-9)
       (* (pow x.re y.re) 1.0)
       (if (<= y.re 0.044) 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 <= -9.5e+101) {
		tmp = t_0;
	} else if (y_46_re <= -2.25e-9) {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	} else if (y_46_re <= 0.044) {
		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 <= (-9.5d+101)) then
        tmp = t_0
    else if (y_46re <= (-2.25d-9)) then
        tmp = (x_46re ** y_46re) * 1.0d0
    else if (y_46re <= 0.044d0) 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 <= -9.5e+101) {
		tmp = t_0;
	} else if (y_46_re <= -2.25e-9) {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	} else if (y_46_re <= 0.044) {
		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 <= -9.5e+101:
		tmp = t_0
	elif y_46_re <= -2.25e-9:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	elif y_46_re <= 0.044:
		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 <= -9.5e+101)
		tmp = t_0;
	elseif (y_46_re <= -2.25e-9)
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	elseif (y_46_re <= 0.044)
		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 <= -9.5e+101)
		tmp = t_0;
	elseif (y_46_re <= -2.25e-9)
		tmp = (x_46_re ^ y_46_re) * 1.0;
	elseif (y_46_re <= 0.044)
		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, -9.5e+101], t$95$0, If[LessEqual[y$46$re, -2.25e-9], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 0.044], 1.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.49999999999999947e101 or 0.043999999999999997 < y.re

   1. Initial program 35.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 63.4%

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

   if -9.49999999999999947e101 < y.re < -2.24999999999999988e-9

   1. Initial program 52.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 62.3%

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

   if -2.24999999999999988e-9 < y.re < 0.043999999999999997

   1. Initial program 41.4%

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

   3. Taylor expanded in y.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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 45.1%

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

Alternative 15: 55.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.5 \cdot 10^{-106}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-27}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -9.5e-106)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 9e-27) (* (pow x.re 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 <= -9.5e-106) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 9e-27) {
		tmp = pow(x_46_re, 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 <= (-9.5d-106)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 9d-27) then
        tmp = (x_46re ** 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 <= -9.5e-106) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 9e-27) {
		tmp = Math.pow(x_46_re, 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 <= -9.5e-106:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 9e-27:
		tmp = math.pow(x_46_re, 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 <= -9.5e-106)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 9e-27)
		tmp = Float64((x_46_re ^ 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 <= -9.5e-106)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 9e-27)
		tmp = (x_46_re ^ 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, -9.5e-106], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 9e-27], N[(N[Power[x$46$re, 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 3 regimes

2. if x.im < -9.4999999999999994e-106

   1. Initial program 39.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in x.im around -inf

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

   11. Applied rewrites 61.2%

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

   if -9.4999999999999994e-106 < x.im < 9.0000000000000003e-27

   1. Initial program 44.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 55.4%

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

   if 9.0000000000000003e-27 < x.im

   1. Initial program 33.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 56.6%

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

Alternative 16: 52.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 0.044:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.2e-12) t_0 (if (<= y.re 0.044) 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 <= -5.2e-12) {
		tmp = t_0;
	} else if (y_46_re <= 0.044) {
		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 <= (-5.2d-12)) then
        tmp = t_0
    else if (y_46re <= 0.044d0) 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 <= -5.2e-12) {
		tmp = t_0;
	} else if (y_46_re <= 0.044) {
		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 <= -5.2e-12:
		tmp = t_0
	elif y_46_re <= 0.044:
		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 <= -5.2e-12)
		tmp = t_0;
	elseif (y_46_re <= 0.044)
		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 <= -5.2e-12)
		tmp = t_0;
	elseif (y_46_re <= 0.044)
		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, -5.2e-12], t$95$0, If[LessEqual[y$46$re, 0.044], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.19999999999999965e-12 or 0.043999999999999997 < y.re

   1. Initial program 37.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 57.7%

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

   if -5.19999999999999965e-12 < y.re < 0.043999999999999997

   1. Initial program 41.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 45.0%

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

Alternative 17: 25.5% 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 39.6%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
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. lower-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-atan2.f64 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 22.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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)))))