Complex division, real part

Average Accuracy: 60.2% → 99.0%

Time: 21.8s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (* (/ 1.0 (hypot c d)) (+ (/ b (/ (hypot c d) d)) (/ a (/ (hypot c d) c)))))

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) {
	return (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / d)) + (a / (hypot(c, d) / c)));
}

Java

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

↓

public static double code(double a, double b, double c, double d) {
	return (1.0 / Math.hypot(c, d)) * ((b / (Math.hypot(c, d) / d)) + (a / (Math.hypot(c, d) / c)));
}

Python

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

↓

def code(a, b, c, d):
	return (1.0 / math.hypot(c, d)) * ((b / (math.hypot(c, d) / d)) + (a / (math.hypot(c, d) / c)))

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)
	return Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(b / Float64(hypot(c, d) / d)) + Float64(a / Float64(hypot(c, d) / c))))
end

MATLAB

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

↓

function tmp = code(a, b, c, d)
	tmp = (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / d)) + (a / (hypot(c, d) / c)));
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_] := N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	60.2%
Target	99.4%
Herbie	99.0%

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

Derivation

1. Initial program 60.2%

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

2. Applied egg-rr 74.2%

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

   [Start] 60.2	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   *-un-lft-identity [=>] 60.2	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
   add-sqr-sqrt [=>] 60.2	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
   times-frac [=>] 60.2	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
   hypot-def [=>] 60.2	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
   fma-def [=>] 60.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
   hypot-def [=>] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

3. Applied egg-rr 74.2%

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

   [Start] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]
   div-inv [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   *-commutative [<=] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, c, b \cdot d\right)\right)} \]
   fma-udef [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a \cdot c + b \cdot d\right)}\right) \]
   distribute-lft-in [=>] 74.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a \cdot c\right) + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right)} \]
   associate-*l/ [=>] 74.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(a \cdot c\right)}{\mathsf{hypot}\left(c, d\right)}} + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right) \]
   *-un-lft-identity [<=] 74.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{\color{blue}{a \cdot c}}{\mathsf{hypot}\left(c, d\right)} + \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b \cdot d\right)\right) \]
   associate-*l/ [=>] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \color{blue}{\frac{1 \cdot \left(b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}\right) \]
   *-un-lft-identity [<=] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{\color{blue}{b \cdot d}}{\mathsf{hypot}\left(c, d\right)}\right) \]

4. Simplified 99.0%

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

   [Start] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
   +-commutative [=>] 74.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right)} \]
   associate-/l* [=>] 85.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right) \]
   associate-/l* [=>] 99.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \color{blue}{\frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}\right) \]

5. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	81.6%
Cost	21088

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{a \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{-110}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 980000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	20352

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

Alternative 3
Accuracy	82.8%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-87}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	82.6%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -9.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + a\right)\\ \end{array} \]

Alternative 5
Accuracy	77.6%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{+109}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 60000000000000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternative 6
Accuracy	77.2%
Cost	7828

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 7
Accuracy	77.3%
Cost	7828

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -2.75 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 4300000000000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternative 8
Accuracy	68.8%
Cost	1497

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4.3 \cdot 10^{-18} \lor \neg \left(d \leq -2.7 \cdot 10^{-52}\right) \land d \leq 440000000000:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 9
Accuracy	68.7%
Cost	1497

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-17} \lor \neg \left(d \leq -3.4 \cdot 10^{-52}\right) \land d \leq 8000000000000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Accuracy	68.5%
Cost	1496

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -59000000000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11
Accuracy	68.5%
Cost	1496

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c}{\frac{d}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 12
Accuracy	74.1%
Cost	1496

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.82 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c}{\frac{d}{c}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	73.6%
Cost	1496

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ \mathbf{elif}\;d \leq 500000000000:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c}{\frac{d}{c}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Accuracy	73.6%
Cost	1496

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -17000000000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -6.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.55 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c}{\frac{d}{c}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Accuracy	77.1%
Cost	1488

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-154}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Accuracy	62.8%
Cost	986

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{+108} \lor \neg \left(d \leq -1.46 \cdot 10^{+15}\right) \land \left(d \leq -9 \cdot 10^{-18} \lor \neg \left(d \leq -9 \cdot 10^{-56}\right) \land d \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 17
Accuracy	41.2%
Cost	324

\[\begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 18
Accuracy	40.5%
Cost	192

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

Reproduce

herbie shell --seed 2023131 
(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))))