_divideComplex, real part

Average Accuracy: 60.2% → 92.9%

Time: 23.5s

Precision: binary64

Cost: 26760

Math

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

↓

\[\begin{array}{l} t_0 := \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 300000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, t_0, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, t_0, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (hypot y.re y.im))))
   (if (<= y.re -1.25e+119)
     (/ (fma -1.0 (* (/ y.im y.re) x.im) (- x.re)) (hypot y.re y.im))
     (if (<= y.re 300000.0)
       (/ (fma x.im t_0 (/ (* y.re x.re) (hypot y.re y.im))) (hypot y.re y.im))
       (/ (fma x.im t_0 x.re) (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -1.25e+119) {
		tmp = fma(-1.0, ((y_46_im / y_46_re) * x_46_im), -x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 300000.0) {
		tmp = fma(x_46_im, t_0, ((y_46_re * x_46_re) / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	} else {
		tmp = fma(x_46_im, t_0, x_46_re) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.25e+119)
		tmp = Float64(fma(-1.0, Float64(Float64(y_46_im / y_46_re) * x_46_im), Float64(-x_46_re)) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 300000.0)
		tmp = Float64(fma(x_46_im, t_0, Float64(Float64(y_46_re * x_46_re) / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(fma(x_46_im, t_0, x_46_re) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

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

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e+119], N[(N[(-1.0 * N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 300000.0], N[(N[(x$46$im * t$95$0 + N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * t$95$0 + x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.25e119

   1. Initial program 35.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Applied egg-rr 56.7%

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

      [Start] 35.8	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 35.8	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 35.8	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 35.8	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 35.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 35.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 56.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      div-inv [=>] 56.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      *-commutative [<=] 56.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      fma-udef [=>] 56.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
      distribute-lft-in [=>] 56.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      *-un-lft-identity [<=] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      associate-*l/ [=>] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
      *-un-lft-identity [<=] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   4. Simplified 64.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      +-commutative [<=] 56.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      associate-/l* [=>] 64.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      *-commutative [=>] 64.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   5. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
      Proof

      [Start] 64.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      expm1-log1p-u [=>] 63.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)\right)} \]
      expm1-udef [=>] 28.8	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} - 1} \]
      associate-/r/ [<=] 28.8	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}}\right)} - 1 \]
      clear-num [<=] 28.8	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]
      div-inv [=>] 28.8	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im \cdot \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      fma-def [=>] 28.8	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      clear-num [<=] 28.8	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      *-commutative [=>] 28.8	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      associate-/l* [=>] 37.0	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      Proof

      [Start] 37.0	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      expm1-def [=>] 90.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]
      expm1-log1p [=>] 99.2	\[ \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      associate-/r/ [=>] 99.2	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 64.2	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   7. Taylor expanded in y.re around -inf 79.9%

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

   8. Simplified 85.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof

      [Start] 79.9	\[ \frac{-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      fma-def [=>] 79.9	\[ \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{y.im \cdot x.im}{y.re}, -1 \cdot x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-/l* [=>] 86.2	\[ \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}, -1 \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-/r/ [=>] 85.5	\[ \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{y.im}{y.re} \cdot x.im}, -1 \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      mul-1-neg [=>] 85.5	\[ \frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, \color{blue}{-x.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -1.25e119 < y.re < 3e5

   1. Initial program 71.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Applied egg-rr 82.7%

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

      [Start] 71.7	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 71.7	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 71.7	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 71.7	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 71.7	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 71.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 82.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      div-inv [=>] 82.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      *-commutative [<=] 82.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      fma-udef [=>] 82.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
      distribute-lft-in [=>] 82.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 82.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      *-un-lft-identity [<=] 82.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      associate-*l/ [=>] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
      *-un-lft-identity [<=] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   4. Simplified 96.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      +-commutative [<=] 82.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      associate-/l* [=>] 96.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      *-commutative [=>] 96.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   5. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
      Proof

      [Start] 96.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      expm1-log1p-u [=>] 71.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)\right)} \]
      expm1-udef [=>] 35.5	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} - 1} \]
      associate-/r/ [<=] 35.5	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}}\right)} - 1 \]
      clear-num [<=] 35.5	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]
      div-inv [=>] 35.5	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im \cdot \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      fma-def [=>] 35.5	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      clear-num [<=] 35.5	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      *-commutative [=>] 35.5	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      associate-/l* [=>] 36.3	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      Proof

      [Start] 36.3	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      expm1-def [=>] 73.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]
      expm1-log1p [=>] 99.2	\[ \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      associate-/r/ [=>] 98.3	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 96.7	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if 3e5 < y.re

   1. Initial program 48.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Applied egg-rr 65.1%

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

      [Start] 48.6	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 48.6	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 48.6	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 48.6	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 48.6	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 48.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 65.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      div-inv [=>] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      *-commutative [<=] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      fma-udef [=>] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
      distribute-lft-in [=>] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      *-un-lft-identity [<=] 65.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      associate-*l/ [=>] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
      *-un-lft-identity [<=] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   4. Simplified 73.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      Proof

      [Start] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      +-commutative [<=] 65.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      associate-/l* [=>] 73.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      *-commutative [=>] 73.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   5. Applied egg-rr 34.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
      Proof

      [Start] 73.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      expm1-log1p-u [=>] 66.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)\right)} \]
      expm1-udef [=>] 27.5	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} - 1} \]
      associate-/r/ [<=] 27.5	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}}\right)} - 1 \]
      clear-num [<=] 27.5	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]
      div-inv [=>] 27.5	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im \cdot \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      fma-def [=>] 27.5	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      clear-num [<=] 27.5	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      *-commutative [=>] 27.5	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      associate-/l* [=>] 34.4	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      Proof

      [Start] 34.4	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]
      expm1-def [=>] 86.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]
      expm1-log1p [=>] 99.6	\[ \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      associate-/r/ [=>] 99.6	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 74.1	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   7. Taylor expanded in y.re around inf 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{x.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 300000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.7%
Cost	21088

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.28 \cdot 10^{-110}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 52000000000:\\ \;\;\;\;\frac{\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	92.7%
Cost	20616

\[\begin{array}{l} t_0 := \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 300000:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot t_0 + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, t_0, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3
Accuracy	83.7%
Cost	20172

\[\begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4
Accuracy	83.5%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{y.im}{y.re} \cdot x.im, -x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 5
Accuracy	78.2%
Cost	13640

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{+109}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1080000000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 6
Accuracy	68.3%
Cost	1499

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135} \lor \neg \left(y.im \leq -1.4 \cdot 10^{+109} \lor \neg \left(y.im \leq -2.9 \cdot 10^{+15}\right) \land \left(y.im \leq -2.65 \cdot 10^{-17} \lor \neg \left(y.im \leq -1.02 \cdot 10^{-79}\right) \land y.im \leq 1.52 \cdot 10^{+15}\right)\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Accuracy	68.5%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 365000000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	68.5%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -13500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.62 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 9
Accuracy	74.8%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -14000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 16000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 10
Accuracy	75.5%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 82000000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 11
Accuracy	77.9%
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 15800000000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 12
Accuracy	62.9%
Cost	986

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{+109} \lor \neg \left(y.im \leq -43000000000000\right) \land \left(y.im \leq -2.55 \cdot 10^{-17} \lor \neg \left(y.im \leq -3.2 \cdot 10^{-52}\right) \land y.im \leq 1.55 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13
Accuracy	42.1%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+201}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 14
Accuracy	41.5%
Cost	192

\[\frac{x.im}{y.im} \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))