_divideComplex, imaginary part

Average Error: 25.8 → 13.5

Time: 24.9s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im + \left(-x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(x.re \cdot y.im + \left(-y.re \cdot x.im\right)\right)\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-\frac{y.im}{y.re}\right) \cdot x.re + x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im}{y.re} + \left(-\frac{y.im}{y.re \cdot y.re} \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re + \left(-x.re\right)}{y.im}\\ \end{array} \]

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 -3.3e+128)
   (/ (+ (* (/ y.re y.im) x.im) (- x.re)) y.im)
   (if (<= y.im -7.8e-114)
     (*
      (/ -1.0 (fma y.im y.im (* y.re y.re)))
      (+ (* x.re y.im) (- (* y.re x.im))))
     (if (<= y.im 5.2e-90)
       (/ (+ (* (- (/ y.im y.re)) x.re) x.im) y.re)
       (if (<= y.im 5.4e-64)
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 1.2e+22)
           (+ (/ x.im y.re) (- (* (/ y.im (* y.re y.re)) x.re)))
           (/ (+ (* (/ x.im y.im) y.re) (- 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 <= -3.3e+128) {
		tmp = (((y_46_re / y_46_im) * x_46_im) + -x_46_re) / y_46_im;
	} else if (y_46_im <= -7.8e-114) {
		tmp = (-1.0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * ((x_46_re * y_46_im) + -(y_46_re * x_46_im));
	} else if (y_46_im <= 5.2e-90) {
		tmp = ((-(y_46_im / y_46_re) * x_46_re) + x_46_im) / y_46_re;
	} else if (y_46_im <= 5.4e-64) {
		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));
	} else if (y_46_im <= 1.2e+22) {
		tmp = (x_46_im / y_46_re) + -((y_46_im / (y_46_re * y_46_re)) * x_46_re);
	} else {
		tmp = (((x_46_im / y_46_im) * y_46_re) + -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 <= -3.3e+128)
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_im) + Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -7.8e-114)
		tmp = Float64(Float64(-1.0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * Float64(Float64(x_46_re * y_46_im) + Float64(-Float64(y_46_re * x_46_im))));
	elseif (y_46_im <= 5.2e-90)
		tmp = Float64(Float64(Float64(Float64(-Float64(y_46_im / y_46_re)) * x_46_re) + x_46_im) / y_46_re);
	elseif (y_46_im <= 5.4e-64)
		tmp = 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)));
	elseif (y_46_im <= 1.2e+22)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(-Float64(Float64(y_46_im / Float64(y_46_re * y_46_re)) * x_46_re)));
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * y_46_re) + 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, -3.3e+128], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.8e-114], N[(N[(-1.0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$im), $MachinePrecision] + (-N[(y$46$re * x$46$im), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.2e-90], N[(N[(N[((-N[(y$46$im / y$46$re), $MachinePrecision]) * x$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.4e-64], 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], If[LessEqual[y$46$im, 1.2e+22], N[(N[(x$46$im / y$46$re), $MachinePrecision] + (-N[(N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.im < -3.3000000000000001e128

   1. Initial program 42.0

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

   2. Simplified 42.0

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

   3. Taylor expanded in y.re around 0 42.4

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

   4. Simplified 42.4

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

   5. Taylor expanded in y.im around inf 13.3

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

   6. Simplified 12.3

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

   7. Applied egg-rr 8.7

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

   if -3.3000000000000001e128 < y.im < -7.80000000000000003e-114

   1. Initial program 17.2

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

   2. Simplified 17.2

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

   3. Applied egg-rr 17.2

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

   4. Simplified 17.2

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

   if -7.80000000000000003e-114 < y.im < 5.2000000000000001e-90

   1. Initial program 21.5

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

   2. Taylor expanded in y.re around inf 12.2

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

   3. Simplified 12.2

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

   4. Applied egg-rr 12.5

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

   5. Simplified 9.4

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

   6. Applied egg-rr 8.5

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

   if 5.2000000000000001e-90 < y.im < 5.39999999999999971e-64

   1. Initial program 14.6

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

   if 5.39999999999999971e-64 < y.im < 1.2e22

   1. Initial program 14.4

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

   2. Taylor expanded in y.re around inf 29.9

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

   3. Simplified 29.9

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

   4. Applied egg-rr 30.1

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

   if 1.2e22 < y.im

   1. Initial program 33.0

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

   2. Simplified 33.0

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

   3. Taylor expanded in y.re around 0 37.6

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

   4. Simplified 37.6

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

   5. Taylor expanded in y.im around inf 17.8

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

   6. Simplified 18.0

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

   7. Applied egg-rr 14.6

      \[\leadsto \frac{\color{blue}{\frac{x.im}{y.im} \cdot y.re} + \left(-x.re\right)}{y.im} \]
3. Recombined 6 regimes into one program.

Alternatives

Alternative 1
Error	13.6
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im + \left(-x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-\frac{y.im}{y.re}\right) \cdot x.re + x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.76 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im}{y.re} + \left(-\frac{y.im}{y.re \cdot y.re} \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re + \left(-x.re\right)}{y.im}\\ \end{array} \]

Alternative 2
Error	17.1
Cost	1168

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -7.6 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(-\frac{y.im}{y.re}\right) \cdot x.re + x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.6
Cost	1168

\[\begin{array}{l} t_0 := \frac{\frac{x.im}{y.im} \cdot y.re + \left(-x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(-\frac{y.im}{y.re}\right) \cdot x.re + x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.5
Cost	1168

\[\begin{array}{l} t_0 := \frac{\frac{x.im}{y.im} \cdot y.re + \left(-x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.re} + \left(-\frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -7.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(-\frac{y.im}{y.re}\right) \cdot x.re + x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.1
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ t_1 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7.6 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.6
Cost	840

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	24.7
Cost	520

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2023010 
(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))))