_divideComplex, imaginary part

Average Error: 41.55% → 19.43%

Time: 12.3s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{x.re} \cdot \frac{y.re}{y.im}}\\ \end{array} \]

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.re -5e+54)
   (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
   (if (<= y.re -1.6e-119)
     (*
      (/ 1.0 (hypot y.re y.im))
      (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im)))
     (if (<= y.re 9.5e+14)
       (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
       (+ (/ x.im y.re) (/ -1.0 (* (/ y.re x.re) (/ y.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_re <= -5e+54) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_re <= -1.6e-119) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 9.5e+14) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / x_46_re) * (y_46_re / y_46_im)));
	}
	return tmp;
}

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) {
	double tmp;
	if (y_46_re <= -5e+54) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_re <= -1.6e-119) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 9.5e+14) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / x_46_re) * (y_46_re / y_46_im)));
	}
	return tmp;
}

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):
	tmp = 0
	if y_46_re <= -5e+54:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_re <= -1.6e-119:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= 9.5e+14:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / x_46_re) * (y_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_re <= -5e+54)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	elseif (y_46_re <= -1.6e-119)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 9.5e+14)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(-1.0 / Float64(Float64(y_46_re / x_46_re) * Float64(y_46_re / y_46_im))));
	end
	return tmp
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_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5e+54)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_re <= -1.6e-119)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 9.5e+14)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / x_46_re) * (y_46_re / y_46_im)));
	end
	tmp_2 = 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$46$re, -5e+54], 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], If[LessEqual[y$46$re, -1.6e-119], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e+14], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(-1.0 / N[(N[(y$46$re / x$46$re), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.00000000000000005e54

   1. Initial program 58.81

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

   2. Taylor expanded in y.re around inf 27.92

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

   3. Simplified 18.6

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

      [Start] 27.92	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 27.92	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 27.92	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 27.92	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 27.92	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 18.6	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   if -5.00000000000000005e54 < y.re < -1.59999999999999997e-119

   1. Initial program 24.37

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

   2. Applied egg-rr 15.32

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

   if -1.59999999999999997e-119 < y.re < 9.5e14

   1. Initial program 31.69

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

   2. Taylor expanded in y.re around 0 24.55

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

   3. Simplified 22.56

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

      [Start] 24.55	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 24.55	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 24.55	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 24.55	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 24.55	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 24.55	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 22.56	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   4. Applied egg-rr 19.94

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

   if 9.5e14 < y.re

   1. Initial program 51.67

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

   2. Taylor expanded in y.re around inf 28.82

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

   3. Simplified 21.66

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

      [Start] 28.82	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 28.82	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 28.82	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 28.82	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 28.82	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 21.66	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   4. Applied egg-rr 21.84

      \[\leadsto \frac{x.im}{y.re} - \color{blue}{\frac{1}{\frac{y.re}{x.re} \cdot \frac{y.re}{y.im}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 19.43

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{x.re} \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.89%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{x.re} \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 2
Error	23.97%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.92 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{x.re} \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 3
Error	23.94%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.92 \cdot 10^{+33} \lor \neg \left(y.re \leq 2.3 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	30.72%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.92 \cdot 10^{+33} \lor \neg \left(y.re \leq 4.8 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	24.15%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.92 \cdot 10^{+33} \lor \neg \left(y.re \leq 1.85 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 6
Error	36.26%
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.92 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 7
Error	58.16%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(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))))