Complex division, real part

Average Accuracy: 55.3% → 76.4%

Time: 15.1s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \end{array} \]

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 -2.7e+58)
     (* (/ d (hypot d c)) (/ b (hypot d c)))
     (if (<= d -9.5e-224)
       t_0
       (if (<= d -3.4e-299)
         (/ (+ a (/ b (/ c d))) c)
         (if (<= d 9.2e+76) t_0 (+ (/ b d) (/ c (/ (* d d) 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)) * (fma(a, c, (d * b)) / hypot(c, d));
	double tmp;
	if (d <= -2.7e+58) {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	} else if (d <= -9.5e-224) {
		tmp = t_0;
	} else if (d <= -3.4e-299) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 9.2e+76) {
		tmp = t_0;
	} else {
		tmp = (b / d) + (c / ((d * d) / 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(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(d * b)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -2.7e+58)
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	elseif (d <= -9.5e-224)
		tmp = t_0;
	elseif (d <= -3.4e-299)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 9.2e+76)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(c / Float64(Float64(d * d) / a)));
	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, -2.7e+58], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -9.5e-224], t$95$0, If[LessEqual[d, -3.4e-299], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 9.2e+76], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(c / N[(N[(d * d), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	55.3%
Target	93.7%
Herbie	76.4%

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

Derivation

1. Split input into 4 regimes

2. if d < -2.7000000000000001e58

   1. Initial program 36.4%

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

   2. Taylor expanded in a around 0 35.0%

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

   3. Applied egg-rr 82.4%

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

      [Start] 35.0	\[ \frac{d \cdot b}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 35.0	\[ \frac{d \cdot b}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 48.6	\[ \color{blue}{\frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}}} \]
      +-commutative [=>] 48.6	\[ \frac{d}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 48.6	\[ \frac{d}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} \]
      +-commutative [=>] 48.6	\[ \frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \]
      hypot-def [=>] 82.4	\[ \frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \]

   if -2.7000000000000001e58 < d < -9.5000000000000003e-224 or -3.3999999999999998e-299 < d < 9.20000000000000005e76

   1. Initial program 73.5%

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

   2. Applied egg-rr 84.8%

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

      [Start] 73.5	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 73.5	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 73.5	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 73.5	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 73.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 73.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 84.8	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -9.5000000000000003e-224 < d < -3.3999999999999998e-299

   1. Initial program 55.0%

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

   2. Simplified 55.0%

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

      [Start] 55.0	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      fma-def [=>] 55.0	\[ \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      fma-def [=>] 55.0	\[ \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   3. Taylor expanded in c around inf 55.0%

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

   4. Simplified 55.0%

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

      [Start] 55.0	\[ \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{{c}^{2}} \]
      unpow2 [=>] 55.0	\[ \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{c \cdot c}} \]

   5. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{0 + \mathsf{fma}\left(a, c, b \cdot d\right) \cdot {c}^{-2}} \]
      Step-by-step derivation

      [Start] 55.0	\[ \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c} \]
      add-log-exp [=>] 6.1	\[ \color{blue}{\log \left(e^{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}}\right)} \]
      *-un-lft-identity [=>] 6.1	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}}\right)} \]
      log-prod [=>] 6.1	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}}\right)} \]
      metadata-eval [=>] 6.1	\[ \color{blue}{0} + \log \left(e^{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}}\right) \]
      add-log-exp [<=] 55.0	\[ 0 + \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}} \]
      div-inv [=>] 54.8	\[ 0 + \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{c \cdot c}} \]
      pow2 [=>] 54.8	\[ 0 + \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{c}^{2}}} \]
      pow-flip [=>] 57.9	\[ 0 + \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \color{blue}{{c}^{\left(-2\right)}} \]
      metadata-eval [=>] 57.9	\[ 0 + \mathsf{fma}\left(a, c, b \cdot d\right) \cdot {c}^{\color{blue}{-2}} \]

   6. Simplified 89.9%

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

      [Start] 57.9	\[ 0 + \mathsf{fma}\left(a, c, b \cdot d\right) \cdot {c}^{-2} \]
      +-lft-identity [=>] 57.9	\[ \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot {c}^{-2}} \]
      *-commutative [<=] 57.9	\[ \color{blue}{{c}^{-2} \cdot \mathsf{fma}\left(a, c, b \cdot d\right)} \]
      metadata-eval [<=] 57.9	\[ {c}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot \mathsf{fma}\left(a, c, b \cdot d\right) \]
      pow-sqr [<=] 57.7	\[ \color{blue}{\left({c}^{-1} \cdot {c}^{-1}\right)} \cdot \mathsf{fma}\left(a, c, b \cdot d\right) \]
      unpow-1 [=>] 57.7	\[ \left(\color{blue}{\frac{1}{c}} \cdot {c}^{-1}\right) \cdot \mathsf{fma}\left(a, c, b \cdot d\right) \]
      unpow-1 [=>] 57.7	\[ \left(\frac{1}{c} \cdot \color{blue}{\frac{1}{c}}\right) \cdot \mathsf{fma}\left(a, c, b \cdot d\right) \]
      associate-*l* [=>] 64.5	\[ \color{blue}{\frac{1}{c} \cdot \left(\frac{1}{c} \cdot \mathsf{fma}\left(a, c, b \cdot d\right)\right)} \]
      associate-*l/ [=>] 64.7	\[ \frac{1}{c} \cdot \color{blue}{\frac{1 \cdot \mathsf{fma}\left(a, c, b \cdot d\right)}{c}} \]
      *-lft-identity [=>] 64.7	\[ \frac{1}{c} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{c} \]
      associate-*l/ [=>] 65.1	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c}}{c}} \]
      associate-*r/ [=>] 65.1	\[ \frac{\color{blue}{\frac{1 \cdot \mathsf{fma}\left(a, c, b \cdot d\right)}{c}}}{c} \]
      associate-*l/ [<=] 65.0	\[ \frac{\color{blue}{\frac{1}{c} \cdot \mathsf{fma}\left(a, c, b \cdot d\right)}}{c} \]
      fma-udef [=>] 65.0	\[ \frac{\frac{1}{c} \cdot \color{blue}{\left(a \cdot c + b \cdot d\right)}}{c} \]
      *-commutative [<=] 65.0	\[ \frac{\frac{1}{c} \cdot \left(a \cdot c + \color{blue}{d \cdot b}\right)}{c} \]
      distribute-rgt-in [=>] 65.0	\[ \frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{1}{c} + \left(d \cdot b\right) \cdot \frac{1}{c}}}{c} \]
      associate-*l* [=>] 89.8	\[ \frac{\color{blue}{a \cdot \left(c \cdot \frac{1}{c}\right)} + \left(d \cdot b\right) \cdot \frac{1}{c}}{c} \]
      rgt-mult-inverse [=>] 89.9	\[ \frac{a \cdot \color{blue}{1} + \left(d \cdot b\right) \cdot \frac{1}{c}}{c} \]
      *-rgt-identity [=>] 89.9	\[ \frac{\color{blue}{a} + \left(d \cdot b\right) \cdot \frac{1}{c}}{c} \]
      associate-*r/ [=>] 89.9	\[ \frac{a + \color{blue}{\frac{\left(d \cdot b\right) \cdot 1}{c}}}{c} \]
      *-rgt-identity [=>] 89.9	\[ \frac{a + \frac{\color{blue}{d \cdot b}}{c}}{c} \]
      *-commutative [=>] 89.9	\[ \frac{a + \frac{\color{blue}{b \cdot d}}{c}}{c} \]
      associate-/l* [=>] 89.9	\[ \frac{a + \color{blue}{\frac{b}{\frac{c}{d}}}}{c} \]

   if 9.20000000000000005e76 < d

   1. Initial program 46.7%

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

   2. Simplified 46.7%

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

      [Start] 46.7	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      fma-def [=>] 46.7	\[ \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      fma-def [=>] 46.7	\[ \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   3. Taylor expanded in c around 0 79.6%

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

   4. Simplified 85.1%

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

      [Start] 79.6	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      unpow2 [=>] 79.6	\[ \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]
      associate-/l* [=>] 85.1	\[ \frac{b}{d} + \color{blue}{\frac{c}{\frac{d \cdot d}{a}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.8%
Cost	20700

\[\begin{array}{l} t_0 := \mathsf{fma}\left(d, b, c \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ t_1 := \frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-140}:\\ \;\;\;\;\left(a + \frac{d \cdot b}{c}\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \end{array} \]

Alternative 2
Accuracy	72.7%
Cost	14168

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.56 \cdot 10^{-139}:\\ \;\;\;\;\left(a + \frac{d \cdot b}{c}\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \end{array} \]

Alternative 3
Accuracy	76.0%
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+85}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 4
Accuracy	71.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{if}\;c \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 5
Accuracy	70.9%
Cost	1232

\[\begin{array}{l} t_0 := \frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 6
Accuracy	70.3%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+68} \lor \neg \left(c \leq -3.7 \cdot 10^{+19} \lor \neg \left(c \leq -2.9 \cdot 10^{-46}\right) \land c \leq 4.2 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \]

Alternative 7
Accuracy	70.7%
Cost	1105

\[\begin{array}{l} t_0 := \frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{if}\;c \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-47} \lor \neg \left(c \leq 4.7 \cdot 10^{+55}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \end{array} \]

Alternative 8
Accuracy	66.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-28} \lor \neg \left(d \leq 9.2 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]

Alternative 9
Accuracy	60.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Accuracy	39.1%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(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))))