Complex division, real part

Average Error: 25.9 → 9.7

Time: 17.6s

Precision: binary64

Cost: 20432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-240}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)}{c}\\ \mathbf{elif}\;d \leq 8.284516120883395 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

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 (/ (/ (fma a c (* d b)) (hypot c d)) (hypot c d))))
   (if (<= d -1.6483411845380726e+80)
     (/ (- (fma (/ c d) a b)) (hypot c d))
     (if (<= d -1e-170)
       t_0
       (if (<= d 1e-240)
         (+ (/ a c) (/ (* (/ d c) (- b (* d (/ a c)))) c))
         (if (<= d 8.284516120883395e+80)
           t_0
           (+ (/ b d) (* (/ c d) (/ a 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 = (fma(a, c, (d * b)) / hypot(c, d)) / hypot(c, d);
	double tmp;
	if (d <= -1.6483411845380726e+80) {
		tmp = -fma((c / d), a, b) / hypot(c, d);
	} else if (d <= -1e-170) {
		tmp = t_0;
	} else if (d <= 1e-240) {
		tmp = (a / c) + (((d / c) * (b - (d * (a / c)))) / c);
	} else if (d <= 8.284516120883395e+80) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / 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(fma(a, c, Float64(d * b)) / hypot(c, d)) / hypot(c, d))
	tmp = 0.0
	if (d <= -1.6483411845380726e+80)
		tmp = Float64(Float64(-fma(Float64(c / d), a, b)) / hypot(c, d));
	elseif (d <= -1e-170)
		tmp = t_0;
	elseif (d <= 1e-240)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(d / c) * Float64(b - Float64(d * Float64(a / c)))) / c));
	elseif (d <= 8.284516120883395e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / 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[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6483411845380726e+80], N[((-N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision]) / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-170], t$95$0, If[LessEqual[d, 1e-240], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(d / c), $MachinePrecision] * N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.284516120883395e+80], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	25.9
Target	0.4
Herbie	9.7

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

Derivation

1. Split input into 4 regimes

2. if d < -1.6483411845380726e80

   1. Initial program 37.5

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

   2. Applied egg-rr 26.4

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

   3. Applied egg-rr 26.3

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

   4. Taylor expanded in d around -inf 15.8

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

   5. Simplified 11.8

      \[\leadsto \frac{\color{blue}{-\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{\mathsf{hypot}\left(c, d\right)} \]
      Proof
      (neg.f64 (fma.f64 (/.f64 c d) a b)): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 c d) a) b))): 1 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) d)) b)): 34 points increase in error, 17 points decrease in error
      (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (/.f64 (*.f64 c a) d)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 b (/.f64 (*.f64 c a) d)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 b) (*.f64 -1 (/.f64 (*.f64 c a) d)))): 0 points increase in error, 0 points decrease in error

   if -1.6483411845380726e80 < d < -9.99999999999999983e-171 or 9.9999999999999997e-241 < d < 8.28451612088339485e80

   1. Initial program 15.8

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

   2. Applied egg-rr 9.9

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

   3. Applied egg-rr 9.8

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

   if -9.99999999999999983e-171 < d < 9.9999999999999997e-241

   1. Initial program 23.9

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

   2. Taylor expanded in c around inf 12.8

      \[\leadsto \color{blue}{\frac{a}{c} + \left(-1 \cdot \frac{a \cdot {d}^{2}}{{c}^{3}} + \frac{d \cdot b}{{c}^{2}}\right)} \]

   3. Simplified 9.7

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c \cdot c} \cdot \left(b - \frac{d \cdot a}{c}\right)} \]
      Proof
      (+.f64 (/.f64 a c) (*.f64 (/.f64 d (*.f64 c c)) (-.f64 b (/.f64 (*.f64 d a) c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (*.f64 (/.f64 d (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (-.f64 b (/.f64 (*.f64 d a) c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (*.f64 (/.f64 d (pow.f64 c 2)) (-.f64 b (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 a d)) c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 b (/.f64 d (pow.f64 c 2))) (*.f64 (/.f64 (*.f64 a d) c) (/.f64 d (pow.f64 c 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 d (pow.f64 c 2)) b)) (*.f64 (/.f64 (*.f64 a d) c) (/.f64 d (pow.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 d b) (pow.f64 c 2))) (*.f64 (/.f64 (*.f64 a d) c) (/.f64 d (pow.f64 c 2))))): 17 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 a d) d) (*.f64 c (pow.f64 c 2)))))): 24 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 a (*.f64 d d))) (*.f64 c (pow.f64 c 2))))): 7 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (/.f64 (*.f64 a (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (*.f64 c (pow.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (/.f64 (*.f64 a (pow.f64 d 2)) (*.f64 c (Rewrite=> unpow2_binary64 (*.f64 c c)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (-.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (/.f64 (*.f64 a (pow.f64 d 2)) (Rewrite<= cube-mult_binary64 (pow.f64 c 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (neg.f64 (/.f64 (*.f64 a (pow.f64 d 2)) (pow.f64 c 3)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (+.f64 (/.f64 (*.f64 d b) (pow.f64 c 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 a (pow.f64 d 2)) (pow.f64 c 3)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a c) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 a (pow.f64 d 2)) (pow.f64 c 3))) (/.f64 (*.f64 d b) (pow.f64 c 2))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 3.8

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

   if 8.28451612088339485e80 < d

   1. Initial program 39.1

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

   2. Taylor expanded in c around 0 25.3

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot b}{{d}^{3}} + \left(\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}\right)} \]

   3. Simplified 21.5

      \[\leadsto \color{blue}{\frac{b}{d} - \frac{c}{d \cdot d} \cdot \left(\frac{b \cdot c}{d} - a\right)} \]
      Proof
      (-.f64 (/.f64 b d) (*.f64 (/.f64 c (*.f64 d d)) (-.f64 (/.f64 (*.f64 b c) d) a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (*.f64 (/.f64 c (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (-.f64 (/.f64 (*.f64 b c) d) a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 c (pow.f64 d 2)) (/.f64 (*.f64 b c) d)) (*.f64 (/.f64 c (pow.f64 d 2)) a)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (*.f64 (/.f64 c (pow.f64 d 2)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 c b)) d)) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c (*.f64 c b)) (*.f64 (pow.f64 d 2) d))) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 18 points increase in error, 5 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c c) b)) (*.f64 (pow.f64 d 2) d)) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 13 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) b) (*.f64 (pow.f64 d 2) d)) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (/.f64 (*.f64 (pow.f64 c 2) b) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 d d)) d)) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (/.f64 (*.f64 (pow.f64 c 2) b) (Rewrite<= unpow3_binary64 (pow.f64 d 3))) (*.f64 (/.f64 c (pow.f64 d 2)) a))): 2 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 b d) (-.f64 (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3)) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) (pow.f64 d 2))))): 9 points increase in error, 6 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (/.f64 b d) (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3))) (/.f64 (*.f64 c a) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 b d) (neg.f64 (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3))))) (/.f64 (*.f64 c a) (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 b d) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3))))) (/.f64 (*.f64 c a) (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3))) (/.f64 b d))) (/.f64 (*.f64 c a) (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) b) (pow.f64 d 3))) (+.f64 (/.f64 b d) (/.f64 (*.f64 c a) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in c around 0 18.1

      \[\leadsto \frac{b}{d} - \color{blue}{-1 \cdot \frac{c \cdot a}{{d}^{2}}} \]

   5. Simplified 11.7

      \[\leadsto \frac{b}{d} - \color{blue}{\frac{a}{d} \cdot \frac{-c}{d}} \]
      Proof
      (*.f64 (/.f64 a d) (/.f64 (neg.f64 c) d)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a d) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 c d)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (neg.f64 (/.f64 c d)) (/.f64 a d))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.f64 c d))) (/.f64 a d)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 c d) (/.f64 a d)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c a) (*.f64 d d)))): 67 points increase in error, 38 points decrease in error
      (*.f64 -1 (/.f64 (*.f64 c a) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 10^{-240}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)}{c}\\ \mathbf{elif}\;d \leq 8.284516120883395 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	13508

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)}{c}\\ \mathbf{elif}\;d \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1760

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ t_2 := \frac{c \cdot a}{t_0}\\ t_3 := \frac{d \cdot b}{t_0}\\ \mathbf{if}\;c \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -1.5321459026106533 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-138}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 8.402121517608031 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 3
Error	20.2
Cost	1760

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{d \cdot b}{t_0}\\ t_2 := \frac{c \cdot a}{t_0}\\ \mathbf{if}\;c \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -1.5321459026106533 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8.402121517608031 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{elif}\;c \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 4
Error	20.5
Cost	1628

\[\begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ t_1 := \frac{d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -1.5321459026106533 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.7 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.735828233196281 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9.046285168634949 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8.43042049547238 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 5
Error	13.6
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d \cdot \left(b - d \cdot \frac{a}{c}\right)}{c}}{c}\\ \mathbf{elif}\;d \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c} \cdot \left(b - d \cdot \frac{a}{c}\right)}{c}\\ \mathbf{elif}\;d \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	24.7
Cost	1108

\[\begin{array}{l} t_0 := \frac{d \cdot b}{c \cdot c}\\ \mathbf{if}\;c \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Error	24.6
Cost	1108

\[\begin{array}{l} t_0 := \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 10
Error	20.5
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{c \cdot a}{d}\right)\\ \mathbf{if}\;d \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 12
Error	58.7
Cost	192

\[\frac{a}{d} \]

Alternative 13
Error	37.3
Cost	192

\[\frac{b}{d} \]

Reproduce

herbie shell --seed 2022294 
(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))))