Complex division, real part

Percentage Accurate: 61.6% → 86.0%

Time: 13.2s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{c}{\frac{d}{a}}\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{d} + \frac{t_1}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

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)) (/ (fma a c (* d b)) (hypot c d))))
        (t_1 (/ c (/ d a))))
   (if (<= d -5.6e+141)
     (+ (/ b d) (/ t_1 d))
     (if (<= d -9.5e-134)
       t_0
       (if (<= d 1.3e-76)
         (+ (/ a c) (/ (* b (/ d c)) c))
         (if (<= d 2.9e+71) t_0 (/ (+ b t_1) (hypot c 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 t_1 = c / (d / a);
	double tmp;
	if (d <= -5.6e+141) {
		tmp = (b / d) + (t_1 / d);
	} else if (d <= -9.5e-134) {
		tmp = t_0;
	} else if (d <= 1.3e-76) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (d <= 2.9e+71) {
		tmp = t_0;
	} else {
		tmp = (b + t_1) / hypot(c, 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)))
	t_1 = Float64(c / Float64(d / a))
	tmp = 0.0
	if (d <= -5.6e+141)
		tmp = Float64(Float64(b / d) + Float64(t_1 / d));
	elseif (d <= -9.5e-134)
		tmp = t_0;
	elseif (d <= 1.3e-76)
		tmp = Float64(Float64(a / c) + Float64(Float64(b * Float64(d / c)) / c));
	elseif (d <= 2.9e+71)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + t_1) / hypot(c, 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]}, Block[{t$95$1 = N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.6e+141], N[(N[(b / d), $MachinePrecision] + N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -9.5e-134], t$95$0, If[LessEqual[d, 1.3e-76], N[(N[(a / c), $MachinePrecision] + N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.9e+71], t$95$0, N[(N[(b + t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	61.6%
Target	99.4%
Herbie	86.0%

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

Derivation

1. Split input into 4 regimes

2. if d < -5.59999999999999982e141

   1. Initial program 24.0%

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

   2. Applied egg-rr 54.0%

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

      [Start] 24.0%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 24.0%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 24.0%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 24.0%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 24.0%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 24.0%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 54.0%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around 0 80.9%

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

   4. Simplified 89.2%

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

      [Start] 80.9%	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      unpow2 [=>] 80.9%	\[ \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 89.2%	\[ \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   5. Applied egg-rr 89.3%

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

      [Start] 89.2%	\[ \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d} \]
      associate-*r/ [=>] 89.3%	\[ \frac{b}{d} + \color{blue}{\frac{\frac{c}{d} \cdot a}{d}} \]

   6. Applied egg-rr 89.3%

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

      [Start] 89.3%	\[ \frac{b}{d} + \frac{\frac{c}{d} \cdot a}{d} \]
      associate-/r/ [<=] 89.3%	\[ \frac{b}{d} + \frac{\color{blue}{\frac{c}{\frac{d}{a}}}}{d} \]

   if -5.59999999999999982e141 < d < -9.5000000000000008e-134 or 1.3e-76 < d < 2.90000000000000007e71

   1. Initial program 80.2%

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

   2. Applied egg-rr 85.3%

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

      [Start] 80.2%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 80.2%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 80.2%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 80.1%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 80.1%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 80.1%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 85.3%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -9.5000000000000008e-134 < d < 1.3e-76

   1. Initial program 68.7%

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

   2. Applied egg-rr 82.6%

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

      [Start] 68.7%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 68.7%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 68.7%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 68.6%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 68.6%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 68.6%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 82.6%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around inf 88.8%

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

   4. Simplified 95.7%

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

      [Start] 88.8%	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      unpow2 [=>] 88.8%	\[ \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 95.7%	\[ \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   5. Applied egg-rr 97.8%

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

      [Start] 95.7%	\[ \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c} \]
      associate-*r/ [=>] 97.8%	\[ \frac{a}{c} + \color{blue}{\frac{\frac{d}{c} \cdot b}{c}} \]

   if 2.90000000000000007e71 < d

   1. Initial program 50.7%

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

   2. Applied egg-rr 64.6%

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

      [Start] 50.7%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 50.7%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 50.7%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 50.5%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 50.5%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 50.5%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 64.6%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around 0 94.2%

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

   4. Applied egg-rr 94.9%

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

      [Start] 94.2%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
      associate-*l/ [=>] 94.6%	\[ \color{blue}{\frac{1 \cdot \left(b + \frac{c \cdot a}{d}\right)}{\mathsf{hypot}\left(c, d\right)}} \]
      *-un-lft-identity [<=] 94.6%	\[ \frac{\color{blue}{b + \frac{c \cdot a}{d}}}{\mathsf{hypot}\left(c, d\right)} \]
      associate-/l* [=>] 94.9%	\[ \frac{b + \color{blue}{\frac{c}{\frac{d}{a}}}}{\mathsf{hypot}\left(c, d\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{c}{\frac{d}{a}}}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.0%
Cost	20560

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{c}{\frac{d}{a}}\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{d} + \frac{t_1}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Accuracy	83.3%
Cost	7568

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{c}{\frac{d}{a}}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{b}{d} + \frac{t_1}{d}\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	7568

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ t_1 := b + \frac{c}{\frac{d}{a}}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+93}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{d}{\frac{t_0}{b}} + \frac{c}{\frac{t_0}{a}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4
Accuracy	83.1%
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{b}{d} + \frac{\frac{c}{\frac{d}{a}}}{d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	69.1%
Cost	1241

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -520000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{+91} \lor \neg \left(d \leq 3.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{a}{d \cdot d}\\ \end{array} \]

Alternative 6
Accuracy	69.1%
Cost	1240

\[\begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+61}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -55000000:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c}{\frac{d}{c}}}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;c \cdot \frac{a}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Accuracy	70.0%
Cost	1240

\[\begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9500000:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+99}:\\ \;\;\;\;c \cdot \frac{a}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8
Accuracy	63.3%
Cost	976

\[\begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;c \cdot \frac{a}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 9
Accuracy	76.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-69} \lor \neg \left(d \leq 1.15 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 10
Accuracy	75.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ \end{array} \]

Alternative 11
Accuracy	75.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \end{array} \]

Alternative 12
Accuracy	75.1%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \end{array} \]

Alternative 13
Accuracy	75.1%
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{c}{\frac{d}{a}}}{d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{d \cdot d}\\ \end{array} \]

Alternative 14
Accuracy	63.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 15
Accuracy	42.6%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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))))