_divideComplex, imaginary part

Average Error: 41.11% → 19.16%

Time: 19.7s

Precision: binary64

Cost: 14940

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_2 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_3 := y.re \cdot x.im - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{+60}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{t_2}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_3}}\\ \mathbf{elif}\;y.re \leq -60000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{t_3}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.re}{\frac{t_1}{x.im}} - y.im \cdot \frac{x.re}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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.re (/ y.im x.im)) x.re) y.im))
        (t_1 (fma y.re y.re (* y.im y.im)))
        (t_2 (/ 1.0 (hypot y.re y.im)))
        (t_3 (- (* y.re x.im) (* x.re y.im))))
   (if (<= y.re -1.4e+138)
     (- (/ x.im y.re) (/ (/ x.re y.re) (/ y.re y.im)))
     (if (<= y.re -1.85e+60)
       (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
       (if (<= y.re -1.05e+37)
         (/ t_2 (/ (hypot y.re y.im) t_3))
         (if (<= y.re -60000000.0)
           t_0
           (if (<= y.re -2.4e-123)
             (/ (/ t_3 (hypot y.re y.im)) (hypot y.re y.im))
             (if (<= y.re 2.65e-108)
               t_0
               (if (<= y.re 6.8e+98)
                 (- (/ y.re (/ t_1 x.im)) (* y.im (/ x.re t_1)))
                 (* t_2 (- x.im (/ (* x.re y.im) y.re))))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * 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_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	double t_1 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_2 = 1.0 / hypot(y_46_re, y_46_im);
	double t_3 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -1.4e+138) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) / (y_46_re / y_46_im));
	} else if (y_46_re <= -1.85e+60) {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.05e+37) {
		tmp = t_2 / (hypot(y_46_re, y_46_im) / t_3);
	} else if (y_46_re <= -60000000.0) {
		tmp = t_0;
	} else if (y_46_re <= -2.4e-123) {
		tmp = (t_3 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 2.65e-108) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e+98) {
		tmp = (y_46_re / (t_1 / x_46_im)) - (y_46_im * (x_46_re / t_1));
	} else {
		tmp = t_2 * (x_46_im - ((x_46_re * y_46_im) / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * 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(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im)
	t_1 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	t_2 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_3 = Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.4e+138)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) / Float64(y_46_re / y_46_im)));
	elseif (y_46_re <= -1.85e+60)
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.05e+37)
		tmp = Float64(t_2 / Float64(hypot(y_46_re, y_46_im) / t_3));
	elseif (y_46_re <= -60000000.0)
		tmp = t_0;
	elseif (y_46_re <= -2.4e-123)
		tmp = Float64(Float64(t_3 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 2.65e-108)
		tmp = t_0;
	elseif (y_46_re <= 6.8e+98)
		tmp = Float64(Float64(y_46_re / Float64(t_1 / x_46_im)) - Float64(y_46_im * Float64(x_46_re / t_1)));
	else
		tmp = Float64(t_2 * Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e+138], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.85e+60], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.05e+37], N[(t$95$2 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -60000000.0], t$95$0, If[LessEqual[y$46$re, -2.4e-123], N[(N[(t$95$3 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.65e-108], t$95$0, If[LessEqual[y$46$re, 6.8e+98], N[(N[(y$46$re / N[(t$95$1 / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(x$46$re / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if y.re < -1.4e138

   1. Initial program 68.52

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

   2. Applied egg-rr 43.66

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

   3. Applied egg-rr 43.6

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

   4. Taylor expanded in y.re around inf 22.41

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

   5. Simplified 20.18

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}} \]
      Proof

      [Start] 22.41	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 22.41	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 22.41	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      associate-/l* [=>] 20.41	\[ \frac{x.im}{y.re} - \color{blue}{\frac{x.re}{\frac{{y.re}^{2}}{y.im}}} \]
      unpow2 [=>] 20.41	\[ \frac{x.im}{y.re} - \frac{x.re}{\frac{\color{blue}{y.re \cdot y.re}}{y.im}} \]
      associate-/r/ [=>] 20.18	\[ \frac{x.im}{y.re} - \color{blue}{\frac{x.re}{y.re \cdot y.re} \cdot y.im} \]
      *-commutative [=>] 20.18	\[ \frac{x.im}{y.re} - \color{blue}{y.im \cdot \frac{x.re}{y.re \cdot y.re}} \]

   6. Applied egg-rr 10.87

      \[\leadsto \frac{x.im}{y.re} - \color{blue}{\frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}} \]

   if -1.4e138 < y.re < -1.84999999999999994e60

   1. Initial program 34.01

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

   2. Taylor expanded in x.im around inf 48.53

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

   3. Applied egg-rr 31.47

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

   if -1.84999999999999994e60 < y.re < -1.0500000000000001e37

   1. Initial program 25.29

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

   2. Applied egg-rr 18.27

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

   3. Applied egg-rr 18.36

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

   if -1.0500000000000001e37 < y.re < -6e7 or -2.4e-123 < y.re < 2.64999999999999994e-108

   1. Initial program 33.04

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

   2. Taylor expanded in y.re around 0 20.93

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   3. Simplified 18.38

      \[\leadsto \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}} \]
      Proof

      [Start] 20.93	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 20.93	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 20.93	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 20.93	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 20.93	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 20.93	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 18.38	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   4. Taylor expanded in x.im around 0 20.93

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   5. Simplified 16.74

      \[\leadsto \color{blue}{\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}} \]
      Proof

      [Start] 20.93	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 20.93	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      *-commutative [=>] 20.93	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im} \]
      unpow2 [=>] 20.93	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} + -1 \cdot \frac{x.re}{y.im} \]
      associate-/l* [=>] 22.19	\[ \color{blue}{\frac{x.im}{\frac{y.im \cdot y.im}{y.re}}} + -1 \cdot \frac{x.re}{y.im} \]
      mul-1-neg [=>] 22.19	\[ \frac{x.im}{\frac{y.im \cdot y.im}{y.re}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      sub-neg [<=] 22.19	\[ \color{blue}{\frac{x.im}{\frac{y.im \cdot y.im}{y.re}} - \frac{x.re}{y.im}} \]
      associate-/l* [<=] 20.93	\[ \color{blue}{\frac{x.im \cdot y.re}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      associate-/r* [=>] 15.34	\[ \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]
      div-sub [<=] 15.34	\[ \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}} \]
      *-commutative [<=] 15.34	\[ \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} - x.re}{y.im} \]
      associate-/l* [=>] 16.74	\[ \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

   if -6e7 < y.re < -2.4e-123

   1. Initial program 24.07

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

   2. Applied egg-rr 16.94

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

   3. Applied egg-rr 16.76

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

   if 2.64999999999999994e-108 < y.re < 6.79999999999999944e98

   1. Initial program 24.56

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

   2. Taylor expanded in x.im around 0 24.56

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}}} \]

   3. Simplified 24.81

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

      [Start] 24.56	\[ \frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      mul-1-neg [=>] 24.56	\[ \frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}}\right)} \]
      unsub-neg [=>] 24.56	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}}} \]
      associate-/l* [=>] 24.48	\[ \color{blue}{\frac{y.re}{\frac{{y.re}^{2} + {y.im}^{2}}{x.im}}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      unpow2 [=>] 24.48	\[ \frac{y.re}{\frac{\color{blue}{y.re \cdot y.re} + {y.im}^{2}}{x.im}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      unpow2 [=>] 24.48	\[ \frac{y.re}{\frac{y.re \cdot y.re + \color{blue}{y.im \cdot y.im}}{x.im}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      fma-udef [<=] 24.48	\[ \frac{y.re}{\frac{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}{x.im}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      associate-/l* [=>] 20.8	\[ \frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - \color{blue}{\frac{x.re}{\frac{{y.re}^{2} + {y.im}^{2}}{y.im}}} \]
      associate-/r/ [=>] 24.81	\[ \frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - \color{blue}{\frac{x.re}{{y.re}^{2} + {y.im}^{2}} \cdot y.im} \]
      unpow2 [=>] 24.81	\[ \frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - \frac{x.re}{\color{blue}{y.re \cdot y.re} + {y.im}^{2}} \cdot y.im \]
      unpow2 [=>] 24.81	\[ \frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - \frac{x.re}{y.re \cdot y.re + \color{blue}{y.im \cdot y.im}} \cdot y.im \]
      fma-udef [<=] 24.81	\[ \frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot y.im \]

   if 6.79999999999999944e98 < y.re

   1. Initial program 62.66

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

   2. Applied egg-rr 43.46

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

   3. Taylor expanded in y.re around inf 21.43

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot \frac{x.re \cdot y.im}{y.re} + x.im\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 19.16

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{+60}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re \cdot x.im - x.re \cdot y.im}}\\ \mathbf{elif}\;y.re \leq -60000000:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.6%
Cost	14556

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -60000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.42 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

Alternative 2
Error	17.78%
Cost	14556

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := y.re \cdot x.im - x.re \cdot y.im\\ t_2 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_3 := \frac{\frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -2 \cdot 10^{+60}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{t_2}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_1}}\\ \mathbf{elif}\;y.re \leq -6400000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

Alternative 3
Error	19.75%
Cost	14168

\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -5.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

Alternative 4
Error	19.68%
Cost	14168

\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+134}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-188}:\\ \;\;\;\;t_1 \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{t_1}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+170}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re \cdot y.im}{y.re}\right)\\ \end{array} \]

Alternative 5
Error	23.3%
Cost	7568

\[\begin{array}{l} t_0 := y.re \cdot \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -1650000000:\\ \;\;\;\;\frac{x.re - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 4.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6
Error	25.63%
Cost	7172

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{\frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.im \leq -125000:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9500000000000:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	25.89%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{\frac{x.re}{y.re} \cdot y.im}{y.re}\\ t_1 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -920:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 98000000000:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	27.22%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ t_1 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	26.99%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{\frac{x.re}{y.re} \cdot y.im}{y.re}\\ t_1 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3700:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	31.47%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 11
Error	27.19%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 12
Error	28.19%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 13
Error	31.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-51} \lor \neg \left(y.im \leq 2.25 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 14
Error	35.73%
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 15
Error	58.22%
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))