Complex division, real part

Average Error: 26.4 → 10.0

Time: 16.1s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{+133}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-200}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (fma a c (* d b)) (hypot c d)))))
   (if (<= d -8.2e+133)
     (* (+ b (/ c (/ d a))) (/ -1.0 (hypot c d)))
     (if (<= d -1.45e-92)
       t_0
       (if (<= d 1.35e-200)
         (+ (/ a c) (/ b (* c (/ c d))))
         (if (<= d 3.9e+136) t_0 (+ (/ b d) (/ (* a (/ c d)) d))))))))

C

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

↓

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (fma(a, c, (d * b)) / hypot(c, d));
	double tmp;
	if (d <= -8.2e+133) {
		tmp = (b + (c / (d / a))) * (-1.0 / hypot(c, d));
	} else if (d <= -1.45e-92) {
		tmp = t_0;
	} else if (d <= 1.35e-200) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (d <= 3.9e+136) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((a * (c / d)) / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * 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(fma(a, c, Float64(d * b)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -8.2e+133)
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -1.45e-92)
		tmp = t_0;
	elseif (d <= 1.35e-200)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (d <= 3.9e+136)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * 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[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.2e+133], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.45e-92], t$95$0, If[LessEqual[d, 1.35e-200], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.9e+136], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	26.4
Target	0.5
Herbie	10.0

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

Derivation

1. Split input into 4 regimes

2. if d < -8.20000000000000008e133

   1. Initial program 42.1

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

   2. Applied egg-rr 28.0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in d around -inf 11.0

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

   4. Simplified 6.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(-1 \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\right)} \]
      Proof

      [Start] 11.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(-1 \cdot b + -1 \cdot \frac{c \cdot a}{d}\right) \]
      distribute-lft-out [=>] 11.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(-1 \cdot \left(b + \frac{c \cdot a}{d}\right)\right)} \]
      associate-/l* [=>] 6.9	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(-1 \cdot \left(b + \color{blue}{\frac{c}{\frac{d}{a}}}\right)\right) \]

   if -8.20000000000000008e133 < d < -1.44999999999999992e-92 or 1.3500000000000001e-200 < d < 3.90000000000000019e136

   1. Initial program 18.2

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

   2. Applied egg-rr 12.3

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   if -1.44999999999999992e-92 < d < 1.3500000000000001e-200

   1. Initial program 22.0

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

   2. Applied egg-rr 34.2

      \[\leadsto \frac{\color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{a \cdot c - b \cdot d} - \frac{{\left(b \cdot d\right)}^{2}}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]

   3. Simplified 34.2

      \[\leadsto \frac{\color{blue}{\frac{{\left(d \cdot b\right)}^{2} - {\left(c \cdot a\right)}^{2}}{d \cdot b - c \cdot a}}}{c \cdot c + d \cdot d} \]
      Proof

      [Start] 34.2	\[ \frac{\frac{{\left(a \cdot c\right)}^{2}}{a \cdot c - b \cdot d} - \frac{{\left(b \cdot d\right)}^{2}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      div-sub [<=] 34.2	\[ \frac{\color{blue}{\frac{{\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [<=] 34.2	\[ \frac{\frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      associate-*r/ [<=] 34.2	\[ \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [<=] 34.2	\[ \frac{\color{blue}{\left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1\right)} \cdot \frac{1}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      *-commutative [=>] 34.2	\[ \frac{\color{blue}{\left(1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)\right)} \cdot \frac{1}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      associate-*l* [=>] 34.2	\[ \frac{\color{blue}{1 \cdot \left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}\right)}}{c \cdot c + d \cdot d} \]
      metadata-eval [<=] 34.2	\[ \frac{\color{blue}{\frac{-1}{-1}} \cdot \left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}\right)}{c \cdot c + d \cdot d} \]
      associate-*r/ [=>] 34.2	\[ \frac{\frac{-1}{-1} \cdot \color{blue}{\frac{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [=>] 34.2	\[ \frac{\frac{-1}{-1} \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      times-frac [<=] 34.2	\[ \frac{\color{blue}{\frac{-1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)}{-1 \cdot \left(a \cdot c - b \cdot d\right)}}}{c \cdot c + d \cdot d} \]
      neg-mul-1 [<=] 34.2	\[ \frac{\frac{-1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)}{\color{blue}{-\left(a \cdot c - b \cdot d\right)}}}{c \cdot c + d \cdot d} \]

   4. Taylor expanded in d around 0 9.7

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

   5. Simplified 11.1

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b}{\frac{c \cdot c}{d}}} \]
      Proof

      [Start] 9.7	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [=>] 9.7	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      associate-/l* [=>] 11.1	\[ \frac{a}{c} + \color{blue}{\frac{b}{\frac{{c}^{2}}{d}}} \]
      unpow2 [=>] 11.1	\[ \frac{a}{c} + \frac{b}{\frac{\color{blue}{c \cdot c}}{d}} \]

   6. Applied egg-rr 7.6

      \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{\frac{c}{d} \cdot c}} \]

   if 3.90000000000000019e136 < d

   1. Initial program 43.9

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

   2. Applied egg-rr 29.4

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in d around -inf 47.5

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

   4. Simplified 47.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(-1 \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\right)} \]
      Proof

      [Start] 47.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(-1 \cdot b + -1 \cdot \frac{c \cdot a}{d}\right) \]
      distribute-lft-out [=>] 47.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(-1 \cdot \left(b + \frac{c \cdot a}{d}\right)\right)} \]
      associate-/l* [=>] 47.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(-1 \cdot \left(b + \color{blue}{\frac{c}{\frac{d}{a}}}\right)\right) \]

   5. Taylor expanded in d around -inf 17.0

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

   6. Simplified 9.1

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}} \]
      Proof

      [Start] 17.0	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      unpow2 [=>] 17.0	\[ \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]
      associate-/r* [=>] 13.5	\[ \frac{b}{d} + \color{blue}{\frac{\frac{c \cdot a}{d}}{d}} \]
      *-commutative [=>] 13.5	\[ \frac{b}{d} + \frac{\frac{\color{blue}{a \cdot c}}{d}}{d} \]
      associate-*r/ [<=] 9.1	\[ \frac{b}{d} + \frac{\color{blue}{a \cdot \frac{c}{d}}}{d} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+133}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-200}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	20304

\[\begin{array}{l} t_0 := \left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{1}{d}\\ \mathbf{if}\;c \leq -3.9 \cdot 10^{+124}:\\ \;\;\;\;\left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.36 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]

Alternative 2
Error	12.8
Cost	13768

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{1}{d}\\ \mathbf{if}\;c \leq -1.66 \cdot 10^{+124}:\\ \;\;\;\;\left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]

Alternative 3
Error	12.8
Cost	7300

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{1}{d}\\ \mathbf{if}\;c \leq -1.66 \cdot 10^{+124}:\\ \;\;\;\;\left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]

Alternative 4
Error	12.2
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{+126}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{1}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \end{array} \]

Alternative 5
Error	21.2
Cost	1240

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 6
Error	21.1
Cost	1240

\[\begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.66 \cdot 10^{+41}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2100:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Error	21.2
Cost	1240

\[\begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3.45 \cdot 10^{+40}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 450000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8
Error	16.2
Cost	1233

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.22 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-10} \lor \neg \left(d \leq 300000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]

Alternative 9
Error	16.1
Cost	1232

\[\begin{array}{l} t_0 := \left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{1}{d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2 \cdot 10^{+56}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 6900000000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.4
Cost	976

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1800:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11
Error	23.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 0.62:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 12
Error	37.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;d \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 13
Error	37.5
Cost	192

\[\frac{a}{c} \]

Reproduce

herbie shell --seed 2023054 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

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

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