Complex division, real part

Average Error: 40.48% → 6.36%

Time: 18.0s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{if}\;c \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t_0 \cdot \left(t_1 - a\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+96}:\\ \;\;\;\;t_0 \cdot \left(t_1 + \frac{c \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 + a\right)\\ \end{array} \]

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))) (t_1 (/ b (/ (hypot c d) d))))
   (if (<= c -1e+34)
     (* t_0 (- t_1 a))
     (if (<= c 6e+96)
       (* t_0 (+ t_1 (/ (* c a) (hypot c d))))
       (* t_0 (+ t_1 a))))))

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);
	double t_1 = b / (hypot(c, d) / d);
	double tmp;
	if (c <= -1e+34) {
		tmp = t_0 * (t_1 - a);
	} else if (c <= 6e+96) {
		tmp = t_0 * (t_1 + ((c * a) / hypot(c, d)));
	} else {
		tmp = t_0 * (t_1 + a);
	}
	return tmp;
}

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) {
	double t_0 = 1.0 / Math.hypot(c, d);
	double t_1 = b / (Math.hypot(c, d) / d);
	double tmp;
	if (c <= -1e+34) {
		tmp = t_0 * (t_1 - a);
	} else if (c <= 6e+96) {
		tmp = t_0 * (t_1 + ((c * a) / Math.hypot(c, d)));
	} else {
		tmp = t_0 * (t_1 + a);
	}
	return tmp;
}

Python

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

↓

def code(a, b, c, d):
	t_0 = 1.0 / math.hypot(c, d)
	t_1 = b / (math.hypot(c, d) / d)
	tmp = 0
	if c <= -1e+34:
		tmp = t_0 * (t_1 - a)
	elif c <= 6e+96:
		tmp = t_0 * (t_1 + ((c * a) / math.hypot(c, d)))
	else:
		tmp = t_0 * (t_1 + a)
	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(1.0 / hypot(c, d))
	t_1 = Float64(b / Float64(hypot(c, d) / d))
	tmp = 0.0
	if (c <= -1e+34)
		tmp = Float64(t_0 * Float64(t_1 - a));
	elseif (c <= 6e+96)
		tmp = Float64(t_0 * Float64(t_1 + Float64(Float64(c * a) / hypot(c, d))));
	else
		tmp = Float64(t_0 * Float64(t_1 + a));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / hypot(c, d);
	t_1 = b / (hypot(c, d) / d);
	tmp = 0.0;
	if (c <= -1e+34)
		tmp = t_0 * (t_1 - a);
	elseif (c <= 6e+96)
		tmp = t_0 * (t_1 + ((c * a) / hypot(c, d)));
	else
		tmp = t_0 * (t_1 + a);
	end
	tmp_2 = 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[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1e+34], N[(t$95$0 * N[(t$95$1 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6e+96], N[(t$95$0 * N[(t$95$1 + N[(N[(c * a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 + a), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	40.48%
Target	0.63%
Herbie	6.36%

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

Derivation

1. Split input into 3 regimes

2. if c < -9.99999999999999946e33

   1. Initial program 53.52

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

   2. Applied egg-rr 37

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

   3. Applied egg-rr 37

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

   4. Simplified 28.19

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

      [Start] 37	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
      +-commutative [=>] 37	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right)} \]
      associate-/l* [=>] 28.19	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right) \]
      *-commutative [<=] 28.19	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \frac{\color{blue}{c \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]

   5. Taylor expanded in c around -inf 11.96

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

   6. Simplified 11.96

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

      [Start] 11.96	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + -1 \cdot a\right) \]
      mul-1-neg [=>] 11.96	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \color{blue}{\left(-a\right)}\right) \]

   if -9.99999999999999946e33 < c < 6.0000000000000001e96

   1. Initial program 28.48

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

   2. Applied egg-rr 16.99

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

   3. Applied egg-rr 16.99

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

   4. Simplified 3.14

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

      [Start] 16.99	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
      +-commutative [=>] 16.99	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right)} \]
      associate-/l* [=>] 3.14	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right) \]
      *-commutative [<=] 3.14	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \frac{\color{blue}{c \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]

   if 6.0000000000000001e96 < c

   1. Initial program 62.4

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

   2. Applied egg-rr 43.02

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

   3. Applied egg-rr 43.02

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

   4. Simplified 35.6

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

      [Start] 43.02	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)} + \frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)}\right) \]
      +-commutative [=>] 43.02	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(\frac{b \cdot d}{\mathsf{hypot}\left(c, d\right)} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right)} \]
      associate-/l* [=>] 35.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}} + \frac{a \cdot c}{\mathsf{hypot}\left(c, d\right)}\right) \]
      *-commutative [<=] 35.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \frac{\color{blue}{c \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right) \]

   5. Taylor expanded in c around inf 9.43

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \color{blue}{a}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.36

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} - a\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + \frac{c \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + a\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6%
Cost	20932

\[\begin{array}{l} \mathbf{if}\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} - a\right)\\ \end{array} \]

Alternative 2
Error	15.39%
Cost	20168

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{if}\;c \leq -6 \cdot 10^{-41}:\\ \;\;\;\;t_0 \cdot \left(t_1 - a\right)\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(a, c, d \cdot b\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 + a\right)\\ \end{array} \]

Alternative 3
Error	17%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(-a\right) - \frac{d}{\frac{c}{b}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}} + a\right)\\ \end{array} \]

Alternative 4
Error	15.37%
Cost	14028

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{if}\;c \leq -2.9 \cdot 10^{-40}:\\ \;\;\;\;t_0 \cdot \left(t_1 - a\right)\\ \mathbf{elif}\;c \leq -1.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 + a\right)\\ \end{array} \]

Alternative 5
Error	19.55%
Cost	7828

\[\begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternative 6
Error	19.35%
Cost	7828

\[\begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternative 7
Error	25.17%
Cost	1628

\[\begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ t_1 := c \cdot c + d \cdot d\\ t_2 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot b}{t_1}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -8.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{c \cdot a}{t_1}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.78%
Cost	1488

\[\begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.41%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+19} \lor \neg \left(c \leq -1.1 \cdot 10^{-66} \lor \neg \left(c \leq -7.4 \cdot 10^{-97}\right) \land c \leq 6.4 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Error	24.59%
Cost	1232

\[\begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ t_1 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;c \leq -3 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	24.62%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+19} \lor \neg \left(c \leq 6.2 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \end{array} \]

Alternative 12
Error	24.47%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+19} \lor \neg \left(c \leq 1.3 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \end{array} \]

Alternative 13
Error	24.11%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{+17} \lor \neg \left(d \leq 6.2 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \end{array} \]

Alternative 14
Error	24.71%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 15
Error	37.77%
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 16
Error	59.19%
Cost	192

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

Reproduce

herbie shell --seed 2023102 
(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))))