powComplex, real part

Average Error: 33.0 → 7.4

Time: 42.4s

Precision: binary64

Cost: 97804

Math

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

↓

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sqrt{{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt[3]{y.im}\right)}^{2}, {\left(y.im \cdot y.im\right)}^{0.16666666666666666}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :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)))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* y.im (atan2 x.im x.re))))))
   (if (<= y.im -7.4e+229)
     (exp (* y.im (- (atan2 x.im x.re))))
     (if (<= y.im -4.2e+84)
       t_0
       (if (<= y.im 7700000.0)
         (*
          (/ (pow (hypot x.re x.im) y.re) (pow (exp (atan2 x.im x.re)) y.im))
          (sqrt
           (pow
            (cos
             (fma
              (* (log (hypot x.im x.re)) (pow (cbrt y.im) 2.0))
              (pow (* y.im y.im) 0.16666666666666666)
              (* (atan2 x.im x.re) y.re)))
            2.0)))
         t_0)))))

C

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

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -7.4e+229) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= -4.2e+84) {
		tmp = t_0;
	} else if (y_46_im <= 7700000.0) {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / pow(exp(atan2(x_46_im, x_46_re)), y_46_im)) * sqrt(pow(cos(fma((log(hypot(x_46_im, x_46_re)) * pow(cbrt(y_46_im), 2.0)), pow((y_46_im * y_46_im), 0.16666666666666666), (atan2(x_46_im, x_46_re) * y_46_re))), 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_im <= -7.4e+229)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_im <= -4.2e+84)
		tmp = t_0;
	elseif (y_46_im <= 7700000.0)
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / (exp(atan(x_46_im, x_46_re)) ^ y_46_im)) * sqrt((cos(fma(Float64(log(hypot(x_46_im, x_46_re)) * (cbrt(y_46_im) ^ 2.0)), (Float64(y_46_im * y_46_im) ^ 0.16666666666666666), Float64(atan(x_46_im, x_46_re) * y_46_re))) ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -7.4e+229], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$im, -4.2e+84], t$95$0, If[LessEqual[y$46$im, 7700000.0], N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[Cos[N[(N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[y$46$im, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(y$46$im * y$46$im), $MachinePrecision], 0.16666666666666666], $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.40000000000000005e229

   1. Initial program 35.8

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

   2. Taylor expanded in y.im around 0 17.9

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

   3. Taylor expanded in y.re around 0 17.9

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

   4. Taylor expanded in y.re around 0 10.0

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

   5. Simplified 10.0

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
      Proof
      (*.f64 y.im (neg.f64 (atan2.f64 x.im x.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y.im (atan2.f64 x.im x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y.im (atan2.f64 x.im x.re)))): 0 points increase in error, 0 points decrease in error

   if -7.40000000000000005e229 < y.im < -4.20000000000000037e84 or 7.7e6 < y.im

   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. Taylor expanded in y.im around 0 17.2

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

   3. Taylor expanded in y.re around 0 17.2

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

   if -4.20000000000000037e84 < y.im < 7.7e6

   1. Initial program 31.4

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

   2. Simplified 2.2

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
      Proof
      (*.f64 (/.f64 (pow.f64 (hypot.f64 x.re x.im) y.re) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (cos.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (cos.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 71 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re))) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (cos.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (exp.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (atan2.f64 x.im x.re) y.im)))) (cos.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 2 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im)))) (cos.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (fma.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))))) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 85 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (fma.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im (Rewrite<= *-commutative_binary64 (*.f64 (atan2.f64 x.im x.re) y.re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 2.6

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

   4. Applied egg-rr 34.0

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sqrt{{\cos \color{blue}{\left(e^{\left(3 \cdot \log \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right)\right) \cdot 0.3333333333333333}\right)}}^{2}} \]

   5. Applied egg-rr 2.6

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

   6. Applied egg-rr 2.7

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sqrt{{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt[3]{y.im}\right)}^{2}, \color{blue}{{\left(y.im \cdot y.im\right)}^{0.16666666666666666}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sqrt{{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt[3]{y.im}\right)}^{2}, {\left(y.im \cdot y.im\right)}^{0.16666666666666666}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	97804

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sqrt{{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left({\left(y.im \cdot y.im\right)}^{0.16666666666666666}\right)}^{2}, \sqrt[3]{y.im}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.4
Cost	97612

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;t_2 \cdot \sqrt{{\cos \left(\mathsf{fma}\left(t_1 \cdot {\left(\sqrt[3]{y.im}\right)}^{2}, \sqrt[3]{y.im}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;t_2 \cdot \sqrt{0.5 + 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(2 \cdot \mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.3
Cost	85072

\[\begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{+63}:\\ \;\;\;\;{t_0}^{\left(-y.im\right)} \cdot \cos \left({\left(\sqrt{y.im \cdot \log x.im}\right)}^{2}\right)\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{t_0}^{y.im}} \cdot \sqrt{0.5 + 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(2 \cdot \mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	7.3
Cost	72272

\[\begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{+63}:\\ \;\;\;\;{t_0}^{\left(-y.im\right)} \cdot \cos \left({\left(\sqrt{y.im \cdot \log x.im}\right)}^{2}\right)\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{t_0}^{y.im}} \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	7.0
Cost	65488

\[\begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{+63}:\\ \;\;\;\;{t_0}^{\left(-y.im\right)} \cdot \cos \left({\left(\sqrt{y.im \cdot \log x.im}\right)}^{2}\right)\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{t_0}^{y.im}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.6
Cost	59144

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := e^{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := {\left(\sqrt[3]{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{3}\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+219}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -0.00040227882250791697:\\ \;\;\;\;t_0 \cdot t_2\\ \mathbf{elif}\;y.im \leq 1.8525839484859087 \cdot 10^{-27}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{t_1}^{y.im}} \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot 2\right)\right)}\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;{t_1}^{\left(-y.im\right)} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	8.1
Cost	52748

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := e^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+229}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.8525839484859087 \cdot 10^{-27}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{t_1}^{y.im}} \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot 2\right)\right)}\\ \mathbf{elif}\;y.im \leq 7700000:\\ \;\;\;\;{t_1}^{\left(-y.im\right)} \cdot {\left(\sqrt[3]{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	5.7
Cost	45960

\[\begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_0}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;y.re \leq -7.195175365582101 \cdot 10^{-13}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + \cos t_2\right) + -1\right)\\ \mathbf{elif}\;y.re \leq 1.0975686458326274 \cdot 10^{-17}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_2\right)\right) \cdot \frac{1}{e^{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	6.2
Cost	40004

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -7.195175365582101 \cdot 10^{-13}:\\ \;\;\;\;t_0 \cdot \left(\left(1 + \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) + -1\right)\\ \mathbf{elif}\;y.re \leq 1.0975686458326274 \cdot 10^{-17}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	6.2
Cost	39748

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -7.195175365582101 \cdot 10^{-13}:\\ \;\;\;\;t_0 \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.0975686458326274 \cdot 10^{-17}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	6.2
Cost	26760

\[\begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -7.195175365582101 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0975686458326274 \cdot 10^{-17}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	9.8
Cost	20172

\[\begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.re \cdot \log \left(-x.re\right) - t_0}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -1 \cdot 10^{-75}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \mathbf{elif}\;x.re \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0}\\ \end{array} \]

Alternative 13
Error	16.8
Cost	20108

\[\begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;x.re \leq -1.55 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -9.6 \cdot 10^{-215}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \mathbf{elif}\;x.re \leq 6.4 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0}\\ \end{array} \]

Alternative 14
Error	15.6
Cost	19844

\[\begin{array}{l} \mathbf{if}\;x.im \leq 1.25 \cdot 10^{-276}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]

Alternative 15
Error	21.2
Cost	13120

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

Alternative 16
Error	38.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022318 
(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)))))