_divideComplex, imaginary part

Average Accuracy: 59.2% → 89.2%

Time: 16.1s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.re}}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

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
         (-
          (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))
          (/ x.re (+ y.im (/ (* y.re y.re) y.im))))))
   (if (<= y.re -1.14e+126)
     (/ (/ x.im (hypot y.im y.re)) (/ (hypot y.im y.re) y.re))
     (if (<= y.re -3e-115)
       t_0
       (if (<= y.re 6.1e-133)
         (- (* (/ x.im y.im) (/ y.re y.im)) (/ x.re y.im))
         (if (<= y.re 2.1e+147)
           t_0
           (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re 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 = (x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)))) - (x_46_re / (y_46_im + ((y_46_re * y_46_re) / y_46_im)));
	double tmp;
	if (y_46_re <= -1.14e+126) {
		tmp = (x_46_im / hypot(y_46_im, y_46_re)) / (hypot(y_46_im, y_46_re) / y_46_re);
	} else if (y_46_re <= -3e-115) {
		tmp = t_0;
	} else if (y_46_re <= 6.1e-133) {
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.1e+147) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / 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(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))) - Float64(x_46_re / Float64(y_46_im + Float64(Float64(y_46_re * y_46_re) / y_46_im))))
	tmp = 0.0
	if (y_46_re <= -1.14e+126)
		tmp = Float64(Float64(x_46_im / hypot(y_46_im, y_46_re)) / Float64(hypot(y_46_im, y_46_re) / y_46_re));
	elseif (y_46_re <= -3e-115)
		tmp = t_0;
	elseif (y_46_re <= 6.1e-133)
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.1e+147)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / 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[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(y$46$im + N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.14e+126], N[(N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3e-115], t$95$0, If[LessEqual[y$46$re, 6.1e-133], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e+147], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.1399999999999999e126

   1. Initial program 34.7%

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

   2. Taylor expanded in x.im around inf 32.8%

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

   3. Simplified 36.4%

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

      [Start] 32.8	\[ \frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} \]
      associate-/l* [=>] 34.9	\[ \color{blue}{\frac{y.re}{\frac{{y.re}^{2} + {y.im}^{2}}{x.im}}} \]
      associate-/r/ [=>] 36.4	\[ \color{blue}{\frac{y.re}{{y.re}^{2} + {y.im}^{2}} \cdot x.im} \]
      +-commutative [=>] 36.4	\[ \frac{y.re}{\color{blue}{{y.im}^{2} + {y.re}^{2}}} \cdot x.im \]
      unpow2 [=>] 36.4	\[ \frac{y.re}{\color{blue}{y.im \cdot y.im} + {y.re}^{2}} \cdot x.im \]
      fma-def [=>] 36.4	\[ \frac{y.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot x.im \]
      unpow2 [=>] 36.4	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot x.im \]

   4. Applied egg-rr 80.3%

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

      [Start] 36.4	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im \]
      *-commutative [=>] 36.4	\[ \color{blue}{x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \]
      clear-num [=>] 36.5	\[ x.im \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re}}} \]
      un-div-inv [=>] 36.5	\[ \color{blue}{\frac{x.im}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re}}} \]
      add-sqr-sqrt [=>] 36.5	\[ \frac{x.im}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{y.re}} \]
      *-un-lft-identity [=>] 36.5	\[ \frac{x.im}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{\color{blue}{1 \cdot y.re}}} \]
      times-frac [=>] 36.5	\[ \frac{x.im}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}}} \]
      associate-/r* [=>] 36.5	\[ \color{blue}{\frac{\frac{x.im}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}}} \]
      associate-/l* [<=] 36.5	\[ \frac{\color{blue}{\frac{x.im \cdot 1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}} \]
      *-commutative [<=] 36.5	\[ \frac{\frac{\color{blue}{1 \cdot x.im}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}} \]
      *-un-lft-identity [<=] 36.5	\[ \frac{\frac{\color{blue}{x.im}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}} \]
      fma-udef [=>] 36.5	\[ \frac{\frac{x.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}} \]
      hypot-def [=>] 36.5	\[ \frac{\frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{y.re}} \]
      fma-udef [=>] 36.5	\[ \frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}}{y.re}} \]
      hypot-def [=>] 80.3	\[ \frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}{y.re}} \]

   if -1.1399999999999999e126 < y.re < -3.0000000000000002e-115 or 6.1000000000000004e-133 < y.re < 2.10000000000000006e147

   1. Initial program 73.2%

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

   2. Taylor expanded in x.im around 0 73.2%

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

   3. Simplified 79.8%

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

      [Start] 73.2	\[ \frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      mul-1-neg [=>] 73.2	\[ \frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}}\right)} \]
      unsub-neg [=>] 73.2	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.re}^{2} + {y.im}^{2}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}}} \]
      associate-/l* [=>] 74.0	\[ \color{blue}{\frac{y.re}{\frac{{y.re}^{2} + {y.im}^{2}}{x.im}}} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      associate-/r/ [=>] 77.5	\[ \color{blue}{\frac{y.re}{{y.re}^{2} + {y.im}^{2}} \cdot x.im} - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      +-commutative [=>] 77.5	\[ \frac{y.re}{\color{blue}{{y.im}^{2} + {y.re}^{2}}} \cdot x.im - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      unpow2 [=>] 77.5	\[ \frac{y.re}{\color{blue}{y.im \cdot y.im} + {y.re}^{2}} \cdot x.im - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      fma-def [=>] 77.5	\[ \frac{y.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot x.im - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      unpow2 [=>] 77.5	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot x.im - \frac{x.re \cdot y.im}{{y.re}^{2} + {y.im}^{2}} \]
      associate-/l* [=>] 79.8	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \color{blue}{\frac{x.re}{\frac{{y.re}^{2} + {y.im}^{2}}{y.im}}} \]
      +-commutative [=>] 79.8	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\frac{\color{blue}{{y.im}^{2} + {y.re}^{2}}}{y.im}} \]
      unpow2 [=>] 79.8	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\frac{\color{blue}{y.im \cdot y.im} + {y.re}^{2}}{y.im}} \]
      fma-def [=>] 79.8	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\frac{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}{y.im}} \]
      unpow2 [=>] 79.8	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\frac{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)}{y.im}} \]

   4. Taylor expanded in y.im around 0 94.7%

      \[\leadsto \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\color{blue}{\frac{{y.re}^{2}}{y.im} + y.im}} \]

   5. Simplified 94.7%

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

      [Start] 94.7	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\frac{{y.re}^{2}}{y.im} + y.im} \]
      +-commutative [=>] 94.7	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{\color{blue}{y.im + \frac{{y.re}^{2}}{y.im}}} \]
      unpow2 [=>] 94.7	\[ \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im - \frac{x.re}{y.im + \frac{\color{blue}{y.re \cdot y.re}}{y.im}} \]

   if -3.0000000000000002e-115 < y.re < 6.1000000000000004e-133

   1. Initial program 64.7%

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

   2. Applied egg-rr 80.5%

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

      [Start] 64.7	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 64.7	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 64.6	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 64.6	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 64.7	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 80.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.re around 0 51.3%

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

   4. Simplified 50.4%

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

      [Start] 51.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re \cdot x.im}{y.im} + -1 \cdot x.re\right) \]
      +-commutative [=>] 51.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.re + \frac{y.re \cdot x.im}{y.im}\right)} \]
      mul-1-neg [=>] 51.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\left(-x.re\right)} + \frac{y.re \cdot x.im}{y.im}\right) \]
      associate-/l* [=>] 50.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(-x.re\right) + \color{blue}{\frac{y.re}{\frac{y.im}{x.im}}}\right) \]

   5. Taylor expanded in y.re around 0 83.3%

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

   6. Simplified 85.3%

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

      [Start] 83.3	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 83.3	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 83.3	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 83.3	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 83.3	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 83.3	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 85.3	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   if 2.10000000000000006e147 < y.re

   1. Initial program 31.4%

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

   2. Taylor expanded in y.re around inf 77.4%

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

   3. Simplified 90.1%

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

      [Start] 77.4	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 77.4	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 77.4	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 77.4	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 77.4	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 90.1	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.re}}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-115}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.2%
Cost	13508

\[\begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 2
Accuracy	90.1%
Cost	8144

\[\begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 3
Accuracy	87.3%
Cost	1872

\[\begin{array}{l} t_0 := \frac{y.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -9.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Accuracy	79.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} - x.re\right)\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 108000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Accuracy	76.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+19} \lor \neg \left(y.im \leq 185000000\right):\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6
Accuracy	76.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} - x.re\right)\\ \mathbf{elif}\;y.im \leq 5700000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	70.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.45 \cdot 10^{+19} \lor \neg \left(y.im \leq 33000000000\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	71.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+19} \lor \neg \left(y.im \leq 310000000000\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	63.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{-46} \lor \neg \left(y.im \leq 39000\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	44.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11
Accuracy	8.5%
Cost	192

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

Alternative 12
Accuracy	41.2%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(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))))