_divideComplex, imaginary part

Average Error: 26.4 → 1.9

Time: 14.9s

Precision: binary64

Cost: 26944

Math

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

↓

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

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
 (-
  (* (/ 1.0 (hypot y.re y.im)) (/ x.im (/ (hypot y.re y.im) y.re)))
  (/ x.re (* (hypot y.re y.im) (/ (hypot y.re y.im) 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) {
	return ((1.0 / hypot(y_46_re, y_46_im)) * (x_46_im / (hypot(y_46_re, y_46_im) / y_46_re))) - (x_46_re / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_im)));
}

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im / (Math.hypot(y_46_re, y_46_im) / y_46_re))) - (x_46_re / (Math.hypot(y_46_re, y_46_im) * (Math.hypot(y_46_re, y_46_im) / y_46_im)));
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im / (math.hypot(y_46_re, y_46_im) / y_46_re))) - (x_46_re / (math.hypot(y_46_re, y_46_im) * (math.hypot(y_46_re, y_46_im) / y_46_im)))

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)
	return Float64(Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) / y_46_re))) - Float64(x_46_re / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(y_46_re, y_46_im) / y_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((1.0 / hypot(y_46_re, y_46_im)) * (x_46_im / (hypot(y_46_re, y_46_im) / y_46_re))) - (x_46_re / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_im)));
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_] := N[(N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.4

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

2. Applied egg-rr 22.7

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

3. Simplified 13.0

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}} \]
   Proof
   (-.f64 (*.f64 (/.f64 1 (hypot.f64 y.re y.im)) (/.f64 x.im (/.f64 (hypot.f64 y.re y.im) y.re))) (/.f64 x.re (/.f64 (pow.f64 (hypot.f64 y.re y.im) 2) y.im))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (/.f64 1 (hypot.f64 y.re y.im)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.re) (hypot.f64 y.re y.im)))) (/.f64 x.re (/.f64 (pow.f64 (hypot.f64 y.re y.im) 2) y.im))): 37 points increase in error, 7 points decrease in error
   (-.f64 (*.f64 (/.f64 1 (hypot.f64 y.re y.im)) (/.f64 (*.f64 x.im y.re) (hypot.f64 y.re y.im))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re y.im) (pow.f64 (hypot.f64 y.re y.im) 2)))): 51 points increase in error, 5 points decrease in error
   (Rewrite=> fma-neg_binary64 (fma.f64 (/.f64 1 (hypot.f64 y.re y.im)) (/.f64 (*.f64 x.im y.re) (hypot.f64 y.re y.im)) (neg.f64 (/.f64 (*.f64 x.re y.im) (pow.f64 (hypot.f64 y.re y.im) 2))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 1.9

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

5. Final simplification 1.9

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

Alternatives

Alternative 1
Error	4.2
Cost	27216

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.2 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 8.4 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.7
Cost	14156

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.22 \cdot 10^{-191}:\\ \;\;\;\;t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.8
Cost	14028

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.15 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.4
Cost	13896

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.4
Cost	1224

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.1
Cost	1096

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	18.9
Cost	1032

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	18.4
Cost	840

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.0
Cost	520

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	37.8
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(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))))