Complex division, imag part

Percentage Accurate: 62.6% → 85.5%

Time: 10.8s

Precision: binary64

Cost: 14288

Math

\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (- (* c b) (* d a)) (hypot c d)))))
   (if (<= d -2.1e+160)
     (/ (- (* (/ c d) b) a) d)
     (if (<= d -4.5e-130)
       t_0
       (if (<= d 1.6e-79)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 3.6e+37) t_0 (/ (- (* c (/ b d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

↓

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	double tmp;
	if (d <= -2.1e+160) {
		tmp = (((c / d) * b) - a) / d;
	} else if (d <= -4.5e-130) {
		tmp = t_0;
	} else if (d <= 1.6e-79) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 3.6e+37) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

↓

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / Math.hypot(c, d)) * (((c * b) - (d * a)) / Math.hypot(c, d));
	double tmp;
	if (d <= -2.1e+160) {
		tmp = (((c / d) * b) - a) / d;
	} else if (d <= -4.5e-130) {
		tmp = t_0;
	} else if (d <= 1.6e-79) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 3.6e+37) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

↓

def code(a, b, c, d):
	t_0 = (1.0 / math.hypot(c, d)) * (((c * b) - (d * a)) / math.hypot(c, d))
	tmp = 0
	if d <= -2.1e+160:
		tmp = (((c / d) * b) - a) / d
	elif d <= -4.5e-130:
		tmp = t_0
	elif d <= 1.6e-79:
		tmp = (b - (a * (d / c))) / c
	elif d <= 3.6e+37:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

↓

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -2.1e+160)
		tmp = Float64(Float64(Float64(Float64(c / d) * b) - a) / d);
	elseif (d <= -4.5e-130)
		tmp = t_0;
	elseif (d <= 1.6e-79)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 3.6e+37)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

↓

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	tmp = 0.0;
	if (d <= -2.1e+160)
		tmp = (((c / d) * b) - a) / d;
	elseif (d <= -4.5e-130)
		tmp = t_0;
	elseif (d <= 1.6e-79)
		tmp = (b - (a * (d / c))) / c;
	elseif (d <= 3.6e+37)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.1e+160], N[(N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -4.5e-130], t$95$0, If[LessEqual[d, 1.6e-79], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.6e+37], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Target

Original	62.6%
Target	99.3%
Herbie	85.5%

\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if d < -2.09999999999999997e160

   1. Initial program 21.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 84.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}} \]
      Step-by-step derivation

      [Start] 84.0%	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 84.0%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 84.0%	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 84.0%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      unpow2 [=>] 84.0%	\[ \frac{c \cdot b}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 96.9%	\[ \color{blue}{\frac{c}{d} \cdot \frac{b}{d}} - \frac{a}{d} \]

   4. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\frac{\frac{c}{d} \cdot b - a}{d}} \]
      Step-by-step derivation

      [Start] 96.9%	\[ \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d} \]
      associate-*r/ [=>] 96.9%	\[ \color{blue}{\frac{\frac{c}{d} \cdot b}{d}} - \frac{a}{d} \]
      sub-div [=>] 96.9%	\[ \color{blue}{\frac{\frac{c}{d} \cdot b - a}{d}} \]

   if -2.09999999999999997e160 < d < -4.5e-130 or 1.59999999999999994e-79 < d < 3.59999999999999998e37

   1. Initial program 77.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}} \]
      Step-by-step derivation

      [Start] 77.0%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 77.0%	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 77.0%	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 77.0%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 77.0%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 86.6%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -4.5e-130 < d < 1.59999999999999994e-79

   1. Initial program 68.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}} \]
      Step-by-step derivation

      [Start] 68.9%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 68.9%	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 68.9%	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 68.8%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 68.8%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 82.9%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around inf 85.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]

   4. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}} \]
      Step-by-step derivation

      [Start] 85.5%	\[ -1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c} \]
      +-commutative [=>] 85.5%	\[ \color{blue}{\frac{b}{c} + -1 \cdot \frac{a \cdot d}{{c}^{2}}} \]
      mul-1-neg [=>] 85.5%	\[ \frac{b}{c} + \color{blue}{\left(-\frac{a \cdot d}{{c}^{2}}\right)} \]
      unsub-neg [=>] 85.5%	\[ \color{blue}{\frac{b}{c} - \frac{a \cdot d}{{c}^{2}}} \]
      unpow2 [=>] 85.5%	\[ \frac{b}{c} - \frac{a \cdot d}{\color{blue}{c \cdot c}} \]
      associate-/r* [=>] 93.5%	\[ \frac{b}{c} - \color{blue}{\frac{\frac{a \cdot d}{c}}{c}} \]
      associate-/l* [=>] 94.5%	\[ \frac{b}{c} - \frac{\color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   5. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{1 \cdot \frac{b - \frac{a}{\frac{c}{d}}}{c}} \]
      Step-by-step derivation

      [Start] 94.5%	\[ \frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c} \]
      *-un-lft-identity [=>] 94.5%	\[ \color{blue}{1 \cdot \left(\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\right)} \]
      sub-div [=>] 95.6%	\[ 1 \cdot \color{blue}{\frac{b - \frac{a}{\frac{c}{d}}}{c}} \]

   6. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]
      Step-by-step derivation

      [Start] 95.6%	\[ 1 \cdot \frac{b - \frac{a}{\frac{c}{d}}}{c} \]
      *-lft-identity [=>] 95.6%	\[ \color{blue}{\frac{b - \frac{a}{\frac{c}{d}}}{c}} \]
      associate-/l* [<=] 94.6%	\[ \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
      associate-*r/ [<=] 95.6%	\[ \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   if 3.59999999999999998e37 < d

   1. Initial program 48.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}} \]
      Step-by-step derivation

      [Start] 48.7%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 48.7%	\[ \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 48.7%	\[ \frac{1 \cdot \left(b \cdot c - a \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 48.7%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 48.7%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 63.3%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around 0 84.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}}} \]

   4. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{c \cdot \frac{b}{d} - a}{d}} \]
      Step-by-step derivation

      [Start] 84.0%	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      neg-mul-1 [<=] 84.0%	\[ \color{blue}{\left(-\frac{a}{d}\right)} + \frac{c \cdot b}{{d}^{2}} \]
      *-commutative [=>] 84.0%	\[ \left(-\frac{a}{d}\right) + \frac{\color{blue}{b \cdot c}}{{d}^{2}} \]
      unpow2 [=>] 84.0%	\[ \left(-\frac{a}{d}\right) + \frac{b \cdot c}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 90.6%	\[ \left(-\frac{a}{d}\right) + \color{blue}{\frac{b}{d} \cdot \frac{c}{d}} \]
      *-commutative [<=] 90.6%	\[ \left(-\frac{a}{d}\right) + \color{blue}{\frac{c}{d} \cdot \frac{b}{d}} \]
      +-commutative [<=] 90.6%	\[ \color{blue}{\frac{c}{d} \cdot \frac{b}{d} + \left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 90.6%	\[ \color{blue}{\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}} \]
      associate-*l/ [=>] 92.2%	\[ \color{blue}{\frac{c \cdot \frac{b}{d}}{d}} - \frac{a}{d} \]
      div-sub [<=] 92.2%	\[ \color{blue}{\frac{c \cdot \frac{b}{d} - a}{d}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.5%
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b - a}{d}\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 2
Accuracy	83.1%
Cost	1488

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 3
Accuracy	71.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+160} \lor \neg \left(d \leq 560000000\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 4
Accuracy	77.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-66} \lor \neg \left(d \leq 2.2 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 5
Accuracy	77.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 6
Accuracy	63.0%
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-66} \lor \neg \left(d \leq 7.2 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 7
Accuracy	42.4%
Cost	192

\[\frac{b}{c} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))