powComplex, real part

Percentage Accurate: 39.8% → 77.0%

Time: 17.8s

Alternatives: 9

Speedup: 4.9×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 39.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+22}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -1.3e-6)
     t_0
     (if (<= y.re 3.8e+22)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (* y.re (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(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -1.3e-6) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+22) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos((y_46_re * 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(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.3e-6)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+22)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e-6], t$95$0, If[LessEqual[y$46$re, 3.8e+22], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * 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 < -1.30000000000000005e-6 or 3.8000000000000004e22 < y.re

   1. Initial program 30.1%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 75.0%

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

   if -1.30000000000000005e-6 < y.re < 3.8000000000000004e22

   1. Initial program 35.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 82.0

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

   8. Applied rewrites 82.0%

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

Alternative 2: 72.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.re}\right)\\ t_3 := y.im \cdot \log x.re\\ t_4 := \sin t\_0 \cdot \sin t\_3\\ t_5 := \cos t\_0 \cdot \cos t\_3\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(t\_2, -y.im, t\_0\right)\right) \cdot e^{-\mathsf{fma}\left(y.re, t\_2, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_5}^{3} - {t\_4}^{3}}{{t\_5}^{2} + \left({t\_4}^{2} + t\_5 \cdot t\_4\right)} \cdot e^{y.re \cdot \log x.re - t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* y.im (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.re)))
        (t_3 (* y.im (log x.re)))
        (t_4 (* (sin t_0) (sin t_3)))
        (t_5 (* (cos t_0) (cos t_3))))
   (if (<= x.re -5e-309)
     (* (cos (fma t_2 (- y.im) t_0)) (exp (- (fma y.re t_2 t_1))))
     (*
      (/
       (- (pow t_5 3.0) (pow t_4 3.0))
       (+ (pow t_5 2.0) (+ (pow t_4 2.0) (* t_5 t_4))))
      (exp (- (* y.re (log x.re)) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = log((-1.0 / x_46_re));
	double t_3 = y_46_im * log(x_46_re);
	double t_4 = sin(t_0) * sin(t_3);
	double t_5 = cos(t_0) * cos(t_3);
	double tmp;
	if (x_46_re <= -5e-309) {
		tmp = cos(fma(t_2, -y_46_im, t_0)) * exp(-fma(y_46_re, t_2, t_1));
	} else {
		tmp = ((pow(t_5, 3.0) - pow(t_4, 3.0)) / (pow(t_5, 2.0) + (pow(t_4, 2.0) + (t_5 * t_4)))) * exp(((y_46_re * log(x_46_re)) - t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-1.0 / x_46_re))
	t_3 = Float64(y_46_im * log(x_46_re))
	t_4 = Float64(sin(t_0) * sin(t_3))
	t_5 = Float64(cos(t_0) * cos(t_3))
	tmp = 0.0
	if (x_46_re <= -5e-309)
		tmp = Float64(cos(fma(t_2, Float64(-y_46_im), t_0)) * exp(Float64(-fma(y_46_re, t_2, t_1))));
	else
		tmp = Float64(Float64(Float64((t_5 ^ 3.0) - (t_4 ^ 3.0)) / Float64((t_5 ^ 2.0) + Float64((t_4 ^ 2.0) + Float64(t_5 * t_4)))) * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -5e-309], N[(N[Cos[N[(t$95$2 * (-y$46$im) + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(y$46$re * t$95$2 + t$95$1), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[t$95$5, 3.0], $MachinePrecision] - N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$5, 2.0], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(t$95$5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.9999999999999995e-309

   1. Initial program 32.1%

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

   3. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 N/A

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

      14. lower-exp.f64 N/A

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

   5. Applied rewrites 76.7%

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

   if -4.9999999999999995e-309 < x.re

   1. Initial program 34.0%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 79.0%

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

Alternative 3: 77.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 820000000:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -1.3e-6)
     t_0
     (if (<= y.re 820000000.0)
       (/ 1.0 (/ 1.0 (* 1.0 (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(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -1.3e-6) {
		tmp = t_0;
	} else if (y_46_re <= 820000000.0) {
		tmp = 1.0 / (1.0 / (1.0 * exp((-y_46_im * 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(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.3e-6)
		tmp = t_0;
	elseif (y_46_re <= 820000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e-6], t$95$0, If[LessEqual[y$46$re, 820000000.0], N[(1.0 / N[(1.0 / N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.30000000000000005e-6 or 8.2e8 < y.re

   1. Initial program 30.5%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 74.4%

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

   if -1.30000000000000005e-6 < y.re < 8.2e8

   1. Initial program 35.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.7

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

   7. Applied rewrites 34.7%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 82.8%

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

   12. Applied rewrites 82.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 820000000:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 820000000:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{1}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -1.3e-6)
     t_0
     (if (<= y.re 820000000.0)
       (/ 1.0 (/ (exp (* y.im (atan2 x.im x.re))) 1.0))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -1.3e-6) {
		tmp = t_0;
	} else if (y_46_re <= 820000000.0) {
		tmp = 1.0 / (exp((y_46_im * atan2(x_46_im, x_46_re))) / 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.3e-6)
		tmp = t_0;
	elseif (y_46_re <= 820000000.0)
		tmp = Float64(1.0 / Float64(exp(Float64(y_46_im * atan(x_46_im, x_46_re))) / 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e-6], t$95$0, If[LessEqual[y$46$re, 820000000.0], N[(1.0 / N[(N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.30000000000000005e-6 or 8.2e8 < y.re

   1. Initial program 30.5%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 74.4%

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

   if -1.30000000000000005e-6 < y.re < 8.2e8

   1. Initial program 35.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.7

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

   7. Applied rewrites 34.7%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 82.8%

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

Alternative 5: 61.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 31:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -2.3e-36)
     t_0
     (if (<= y.re 31.0) (/ 1.0 (fma (atan2 x.im x.re) y.im 1.0)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -2.3e-36) {
		tmp = t_0;
	} else if (y_46_re <= 31.0) {
		tmp = 1.0 / fma(atan2(x_46_im, x_46_re), y_46_im, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.3e-36)
		tmp = t_0;
	elseif (y_46_re <= 31.0)
		tmp = Float64(1.0 / fma(atan(x_46_im, x_46_re), y_46_im, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e-36], t$95$0, If[LessEqual[y$46$re, 31.0], N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.29999999999999996e-36 or 31 < y.re

   1. Initial program 30.4%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 72.8%

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

   if -2.29999999999999996e-36 < y.re < 31

   1. Initial program 35.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 50.7%

       \[\leadsto \frac{1}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 50.7%

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

Alternative 6: 47.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (/ (* 0.5 (* x.im x.im)) x.re) y.re))))
   (if (<= y.re -2.3e-36)
     t_0
     (if (<= y.re 1.75e-9) (/ 1.0 (fma (atan2 x.im x.re) y.im 1.0)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(((0.5 * (x_46_im * x_46_im)) / x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.3e-36) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e-9) {
		tmp = 1.0 / fma(atan2(x_46_im, x_46_re), y_46_im, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (Float64(Float64(0.5 * Float64(x_46_im * x_46_im)) / x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.3e-36)
		tmp = t_0;
	elseif (y_46_re <= 1.75e-9)
		tmp = Float64(1.0 / fma(atan(x_46_im, x_46_re), y_46_im, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[(N[(0.5 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e-36], t$95$0, If[LessEqual[y$46$re, 1.75e-9], N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.29999999999999996e-36 or 1.75e-9 < y.re

   1. Initial program 30.4%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 45.6%

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

   12. Taylor expanded in y.re around 0

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

   14. Applied rewrites 48.5%

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

   if -2.29999999999999996e-36 < y.re < 1.75e-9

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 51.9%

       \[\leadsto \frac{1}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.9%

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

Alternative 7: 32.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 1.02e+21)
   (/ 1.0 (fma (atan2 x.im x.re) y.im 1.0))
   (fma y.re (log (sqrt (fma x.im x.im (* x.re x.re)))) 1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 1.02e+21) {
		tmp = 1.0 / fma(atan2(x_46_im, x_46_re), y_46_im, 1.0);
	} else {
		tmp = fma(y_46_re, log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))), 1.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 1.02e+21)
		tmp = Float64(1.0 / fma(atan(x_46_im, x_46_re), y_46_im, 1.0));
	else
		tmp = fma(y_46_re, log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_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, 1.02e+21], N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 1.0), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.02e21

   1. Initial program 34.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 33.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 32.5%

       \[\leadsto \frac{1}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 32.5%

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

   if 1.02e21 < y.re

   1. Initial program 28.8%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 37.2%

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

Alternative 8: 26.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 1\right)} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ 1.0 (fma (atan2 x.im x.re) y.im 1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 / fma(atan2(x_46_im, x_46_re), y_46_im, 1.0);
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 / fma(atan(x_46_im, x_46_re), y_46_im, 1.0))
end

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-exp.f64 N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

4. Applied rewrites 30.6%

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

5. Taylor expanded in y.re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 21.0

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

7. Applied rewrites 21.0%

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

8. Taylor expanded in y.im around 0

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

10. Applied rewrites 25.7%

    \[\leadsto \frac{1}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
11. Step-by-step derivation

12. Applied rewrites 25.7%

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

Alternative 9: 26.1% 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 33.0%

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

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 25.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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)))))