_divideComplex, real part

Average Error: 26.5 → 11.7

Time: 22.4s

Precision: binary64

Cost: 20956

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{t_0}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_0 \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-155}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 8.79419289432676 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y.im \cdot \frac{x.im}{y.re} + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, x.re\right)\right)\\ \end{array} \]

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 (/ 1.0 (hypot y.re y.im)))
        (t_1 (/ t_0 (/ (hypot y.re y.im) (fma x.re y.re (* y.im x.im))))))
   (if (<= y.re -4.490116382232771e+114)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (if (<= y.re -5.612379604288589e+70)
       (+ (/ x.im y.im) (/ (/ y.re y.im) (/ y.im x.re)))
       (if (<= y.re -4.567322487019083e+29)
         (* t_0 (/ (+ (* y.re x.re) (* y.im x.im)) (hypot y.re y.im)))
         (if (<= y.re -261.0441113546419)
           (+ (/ x.im y.im) (/ y.re (/ y.im (/ x.re y.im))))
           (if (<= y.re -7.860635256967511e-91)
             t_1
             (if (<= y.re 1e-155)
               (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
               (if (<= y.re 8.79419289432676e+153)
                 t_1
                 (*
                  t_0
                  (+
                   (* y.im (/ x.im y.re))
                   (fma
                    -0.5
                    (* (/ x.re y.re) (/ y.im (/ y.re y.im)))
                    x.re))))))))))))

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 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = t_0 / (hypot(y_46_re, y_46_im) / fma(x_46_re, y_46_re, (y_46_im * x_46_im)));
	double tmp;
	if (y_46_re <= -4.490116382232771e+114) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= -5.612379604288589e+70) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	} else if (y_46_re <= -4.567322487019083e+29) {
		tmp = t_0 * (((y_46_re * x_46_re) + (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -261.0441113546419) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im / (x_46_re / y_46_im)));
	} else if (y_46_re <= -7.860635256967511e-91) {
		tmp = t_1;
	} else if (y_46_re <= 1e-155) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 8.79419289432676e+153) {
		tmp = t_1;
	} else {
		tmp = t_0 * ((y_46_im * (x_46_im / y_46_re)) + fma(-0.5, ((x_46_re / y_46_re) * (y_46_im / (y_46_re / y_46_im))), x_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_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(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(t_0 / Float64(hypot(y_46_re, y_46_im) / fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im))))
	tmp = 0.0
	if (y_46_re <= -4.490116382232771e+114)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= -5.612379604288589e+70)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)));
	elseif (y_46_re <= -4.567322487019083e+29)
		tmp = Float64(t_0 * Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -261.0441113546419)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im / Float64(x_46_re / y_46_im))));
	elseif (y_46_re <= -7.860635256967511e-91)
		tmp = t_1;
	elseif (y_46_re <= 1e-155)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 8.79419289432676e+153)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) + fma(-0.5, Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / Float64(y_46_re / y_46_im))), x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.490116382232771e+114], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.612379604288589e+70], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.567322487019083e+29], N[(t$95$0 * N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -261.0441113546419], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im / N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.860635256967511e-91], t$95$1, If[LessEqual[y$46$re, 1e-155], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.79419289432676e+153], t$95$1, N[(t$95$0 * N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if y.re < -4.490116382232771e114

   1. Initial program 40.3

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

   2. Simplified 40.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 25.5

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

   4. Taylor expanded in y.re around inf 14.8

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

   5. Simplified 8.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 47 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   if -4.490116382232771e114 < y.re < -5.61237960428858887e70

   1. Initial program 20.0

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

   2. Simplified 20.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 16.6

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

   4. Taylor expanded in y.re around 0 45.5

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

   5. Simplified 41.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 40 points increase in error, 17 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 41.4

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

   if -5.61237960428858887e70 < y.re < -4.5673224870190831e29

   1. Initial program 20.4

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

   2. Simplified 20.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 13.7

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

   4. Taylor expanded in x.re around 0 13.7

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

   if -4.5673224870190831e29 < y.re < -261.04411135464193

   1. Initial program 15.6

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

   2. Simplified 15.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 9.0

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

   4. Taylor expanded in y.re around 0 38.9

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

   5. Simplified 35.8

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 40 points increase in error, 17 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 36.1

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

   if -261.04411135464193 < y.re < -7.86063525696751139e-91 or 1.00000000000000001e-155 < y.re < 8.7941928943267593e153

   1. Initial program 17.4

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

   2. Simplified 17.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 11.7

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

   4. Applied egg-rr 11.8

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

   if -7.86063525696751139e-91 < y.re < 1.00000000000000001e-155

   1. Initial program 23.4

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

   2. Simplified 23.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 12.9

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

   4. Taylor expanded in y.re around 0 11.0

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

   5. Simplified 9.8

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 40 points increase in error, 17 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 8.7

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

   if 8.7941928943267593e153 < y.re

   1. Initial program 44.6

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

   2. Simplified 44.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 29.4

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

   4. Taylor expanded in y.re around inf 20.5

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\right)} \]

   5. Simplified 7.8

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{y.re} \cdot y.im + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, x.re\right)\right)} \]
      Proof
      (+.f64 (*.f64 (/.f64 x.im y.re) y.im) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 y.re y.im))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 x.im (/.f64 y.re y.im))) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 y.re y.im))) x.re)): 10 points increase in error, 14 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) y.re)) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 y.re y.im))) x.re)): 19 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) y.re) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 y.re y.im))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) y.re) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im y.im) y.re))) x.re)): 15 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) y.re) (fma.f64 -1/2 (*.f64 (/.f64 x.re y.re) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) y.re)) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) y.re) (fma.f64 -1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (*.f64 y.re y.re))) x.re)): 20 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) y.re) (fma.f64 -1/2 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) y.re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 2))) x.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 2))) x.re) (/.f64 (*.f64 y.im x.im) y.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 2))) (+.f64 x.re (/.f64 (*.f64 y.im x.im) y.re)))): 0 points increase in error, 0 points decrease in error
3. Recombined 7 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{elif}\;y.re \leq 10^{-155}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 8.79419289432676 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.im}{y.re} + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, x.re\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	11.7
Cost	15132

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 8.79419289432676 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y.im \cdot \frac{x.im}{y.re} + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, x.re\right)\right)\\ \end{array} \]

Alternative 2
Error	11.6
Cost	14684

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 8.79419289432676 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.4
Cost	14164

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	17.4
Cost	1620

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ t_1 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -7.860635256967511 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	18.9
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -3.226352485438223 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.9
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.612379604288589 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -261.0441113546419:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -3.226352485438223 \cdot 10^{-47}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	22.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -9.96694085531125 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.226352485438223 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	22.7
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -9.96694085531125 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -3.226352485438223 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Error	24.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -8.504769004369532 \cdot 10^{-72}:\\ \;\;\;\;x.re \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	25.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.226352485438223 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(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))))