_divideComplex, imaginary part

Average Error: 26.8 → 8.2

Time: 22.1s

Precision: binary64

Cost: 33288

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.449340246179198 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re - \frac{y.re}{\frac{y.im}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3446669583345614 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

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
 (if (<= y.im -5.449340246179198e+92)
   (/ (- x.re (/ y.re (/ y.im x.im))) (hypot y.re y.im))
   (if (<= y.im 2.3446669583345614e+166)
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (* y.im (- x.re)) (pow (hypot y.re y.im) 2.0)))
     (- (/ y.re (* y.im (/ y.im x.im))) (/ x.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_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 tmp;
	if (y_46_im <= -5.449340246179198e+92) {
		tmp = (x_46_re - (y_46_re / (y_46_im / x_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 2.3446669583345614e+166) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((y_46_im * -x_46_re) / pow(hypot(y_46_re, y_46_im), 2.0)));
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_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_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)
	tmp = 0.0
	if (y_46_im <= -5.449340246179198e+92)
		tmp = Float64(Float64(x_46_re - Float64(y_46_re / Float64(y_46_im / x_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 2.3446669583345614e+166)
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(y_46_im * Float64(-x_46_re)) / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_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$46$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_] := If[LessEqual[y$46$im, -5.449340246179198e+92], N[(N[(x$46$re - N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3446669583345614e+166], 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] + N[(N[(y$46$im * (-x$46$re)), $MachinePrecision] / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.44934024617919805e92

   1. Initial program 38.8

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

   2. Applied egg-rr 38.8

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

   3. Applied egg-rr 25.8

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

   4. Applied egg-rr 25.7

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

   5. Taylor expanded in y.im around -inf 14.0

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

   6. Simplified 10.6

      \[\leadsto \frac{\color{blue}{x.re - \frac{y.re}{\frac{y.im}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof
      (-.f64 x.re (/.f64 y.re (/.f64 y.im x.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 x.re (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.im) y.im))): 19 points increase in error, 15 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 x.re (neg.f64 (/.f64 (*.f64 y.re x.im) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y.re x.im) y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y.re x.im) y.im)) x.re)): 0 points increase in error, 0 points decrease in error

   if -5.44934024617919805e92 < y.im < 2.3446669583345614e166

   1. Initial program 20.3

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

   2. Applied egg-rr 7.5

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

   if 2.3446669583345614e166 < y.im

   1. Initial program 44.5

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

   2. Applied egg-rr 44.5

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

   3. Applied egg-rr 29.2

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

   4. Taylor expanded in y.re around 0 14.7

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

   5. Simplified 8.5

      \[\leadsto \color{blue}{\frac{y.re}{\frac{y.im}{x.im} \cdot y.im} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (/.f64 y.re (*.f64 (/.f64 y.im x.im) y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y.re (Rewrite<= associate-/r/_binary64 (/.f64 y.im (/.f64 x.im y.im)))) (/.f64 x.re y.im)): 8 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 y.re (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im y.im) x.im))) (/.f64 x.re y.im)): 26 points increase in error, 7 points decrease in error
      (-.f64 (/.f64 y.re (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) x.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2))) (/.f64 x.re y.im)): 17 points increase in error, 12 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (neg.f64 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x.re y.im)) (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.449340246179198 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re - \frac{y.re}{\frac{y.im}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3446669583345614 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	14296

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ t_2 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -8.079571550312491 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re - \frac{y.re}{\frac{y.im}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.649780066403465 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.1775901236353118 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.592457228449449 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -8.079571550312491 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re - \frac{y.re}{\frac{y.im}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-140}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	7172

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -8.079571550312491 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re - \frac{y.re}{\frac{y.im}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.649780066403465 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.1775901236353118 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.592457228449449 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	12.6
Cost	1752

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -8.079571550312491 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.649780066403465 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.1775901236353118 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.592457228449449 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	15.5
Cost	968

\[\begin{array}{l} t_0 := \frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.890787294922116 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.7254142569083375 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.7254142569083375 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	18.5
Cost	840

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.200908651103965 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.8
Cost	520

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.890787294922116 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.200908651103965 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	58.9
Cost	192

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

Alternative 10
Error	36.8
Cost	192

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

Reproduce

herbie shell --seed 2022318 
(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))))