Complex division, imag part

Average Error: 25.9 → 10.5

Time: 13.9s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+59}:\\ \;\;\;\;t_1 \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-185}:\\ \;\;\;\;t_1 \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{-121}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;b \cdot \frac{c}{t_0} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))) (t_1 (/ 1.0 (hypot c d))))
   (if (<= c -1.25e+59)
     (* t_1 (- (/ a (/ c d)) b))
     (if (<= c -8e-185)
       (* t_1 (/ (- (* c b) (* d a)) (hypot c d)))
       (if (<= c 6.4e-121)
         (- (* b (/ (/ c d) d)) (/ a d))
         (if (<= c 6.4e+69)
           (- (* b (/ c t_0)) (/ a (/ t_0 d)))
           (* (/ c (hypot c d)) (/ b (hypot c d)))))))))

C

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

↓

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double t_1 = 1.0 / hypot(c, d);
	double tmp;
	if (c <= -1.25e+59) {
		tmp = t_1 * ((a / (c / d)) - b);
	} else if (c <= -8e-185) {
		tmp = t_1 * (((c * b) - (d * a)) / hypot(c, d));
	} else if (c <= 6.4e-121) {
		tmp = (b * ((c / d) / d)) - (a / d);
	} else if (c <= 6.4e+69) {
		tmp = (b * (c / t_0)) - (a / (t_0 / d));
	} else {
		tmp = (c / hypot(c, d)) * (b / hypot(c, d));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double t_1 = 1.0 / Math.hypot(c, d);
	double tmp;
	if (c <= -1.25e+59) {
		tmp = t_1 * ((a / (c / d)) - b);
	} else if (c <= -8e-185) {
		tmp = t_1 * (((c * b) - (d * a)) / Math.hypot(c, d));
	} else if (c <= 6.4e-121) {
		tmp = (b * ((c / d) / d)) - (a / d);
	} else if (c <= 6.4e+69) {
		tmp = (b * (c / t_0)) - (a / (t_0 / d));
	} else {
		tmp = (c / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	}
	return tmp;
}

Python

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

↓

def code(a, b, c, d):
	t_0 = (c * c) + (d * d)
	t_1 = 1.0 / math.hypot(c, d)
	tmp = 0
	if c <= -1.25e+59:
		tmp = t_1 * ((a / (c / d)) - b)
	elif c <= -8e-185:
		tmp = t_1 * (((c * b) - (d * a)) / math.hypot(c, d))
	elif c <= 6.4e-121:
		tmp = (b * ((c / d) / d)) - (a / d)
	elif c <= 6.4e+69:
		tmp = (b * (c / t_0)) - (a / (t_0 / d))
	else:
		tmp = (c / math.hypot(c, d)) * (b / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

↓

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	t_1 = Float64(1.0 / hypot(c, d))
	tmp = 0.0
	if (c <= -1.25e+59)
		tmp = Float64(t_1 * Float64(Float64(a / Float64(c / d)) - b));
	elseif (c <= -8e-185)
		tmp = Float64(t_1 * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)));
	elseif (c <= 6.4e-121)
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	elseif (c <= 6.4e+69)
		tmp = Float64(Float64(b * Float64(c / t_0)) - Float64(a / Float64(t_0 / d)));
	else
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(b / hypot(c, d)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c, d)
	t_0 = (c * c) + (d * d);
	t_1 = 1.0 / hypot(c, d);
	tmp = 0.0;
	if (c <= -1.25e+59)
		tmp = t_1 * ((a / (c / d)) - b);
	elseif (c <= -8e-185)
		tmp = t_1 * (((c * b) - (d * a)) / hypot(c, d));
	elseif (c <= 6.4e-121)
		tmp = (b * ((c / d) / d)) - (a / d);
	elseif (c <= 6.4e+69)
		tmp = (b * (c / t_0)) - (a / (t_0 / d));
	else
		tmp = (c / hypot(c, d)) * (b / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * 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[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+59], N[(t$95$1 * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -8e-185], N[(t$95$1 * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.4e-121], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.4e+69], N[(N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(a / N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	25.9
Target	0.4
Herbie	10.5

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

Derivation

1. Split input into 5 regimes

2. if c < -1.2499999999999999e59

   1. Initial program 35.4

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

   2. Applied egg-rr 24.6

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

   3. Taylor expanded in c around -inf 14.4

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

   4. Simplified 11.3

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

      [Start] 14.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a \cdot d}{c} + -1 \cdot b\right) \]
      associate-/l* [=>] 11.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{\frac{a}{\frac{c}{d}}} + -1 \cdot b\right) \]
      mul-1-neg [=>] 11.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} + \color{blue}{\left(-b\right)}\right) \]

   if -1.2499999999999999e59 < c < -7.9999999999999999e-185

   1. Initial program 14.2

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

   2. Applied egg-rr 9.5

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

   if -7.9999999999999999e-185 < c < 6.40000000000000038e-121

   1. Initial program 22.6

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

   2. Applied egg-rr 22.8

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

   3. Simplified 22.8

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

      [Start] 22.8	\[ \frac{\left(b \cdot c - a \cdot d\right) + \left(\left(\mathsf{fma}\left(-d, a, a \cdot d\right) + \mathsf{fma}\left(-d, a, a \cdot d\right)\right) + \left(\mathsf{fma}\left(-d, a, a \cdot d\right) + \mathsf{fma}\left(-d, a, a \cdot d\right)\right)\right)}{c \cdot c + d \cdot d} \]
      *-commutative [=>] 22.8	\[ \frac{\left(\color{blue}{c \cdot b} - a \cdot d\right) + \left(\left(\mathsf{fma}\left(-d, a, a \cdot d\right) + \mathsf{fma}\left(-d, a, a \cdot d\right)\right) + \left(\mathsf{fma}\left(-d, a, a \cdot d\right) + \mathsf{fma}\left(-d, a, a \cdot d\right)\right)\right)}{c \cdot c + d \cdot d} \]
      count-2 [=>] 22.8	\[ \frac{\left(c \cdot b - a \cdot d\right) + \color{blue}{2 \cdot \left(\mathsf{fma}\left(-d, a, a \cdot d\right) + \mathsf{fma}\left(-d, a, a \cdot d\right)\right)}}{c \cdot c + d \cdot d} \]
      count-2 [=>] 22.8	\[ \frac{\left(c \cdot b - a \cdot d\right) + 2 \cdot \color{blue}{\left(2 \cdot \mathsf{fma}\left(-d, a, a \cdot d\right)\right)}}{c \cdot c + d \cdot d} \]

   4. Taylor expanded in d around inf 9.1

      \[\leadsto \color{blue}{\left(4 \cdot \frac{a + -1 \cdot a}{d} + \frac{c \cdot b}{{d}^{2}}\right) - \frac{a}{d}} \]

   5. Simplified 6.1

      \[\leadsto \color{blue}{b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}} \]
      Proof

      [Start] 9.1	\[ \left(4 \cdot \frac{a + -1 \cdot a}{d} + \frac{c \cdot b}{{d}^{2}}\right) - \frac{a}{d} \]
      +-commutative [=>] 9.1	\[ \color{blue}{\left(\frac{c \cdot b}{{d}^{2}} + 4 \cdot \frac{a + -1 \cdot a}{d}\right)} - \frac{a}{d} \]
      associate--l+ [=>] 9.1	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + \left(4 \cdot \frac{a + -1 \cdot a}{d} - \frac{a}{d}\right)} \]
      associate-*r/ [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \left(\color{blue}{\frac{4 \cdot \left(a + -1 \cdot a\right)}{d}} - \frac{a}{d}\right) \]
      div-sub [<=] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\frac{4 \cdot \left(a + -1 \cdot a\right) - a}{d}} \]
      distribute-rgt1-in [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \frac{4 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot a\right)} - a}{d} \]
      metadata-eval [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \frac{4 \cdot \left(\color{blue}{0} \cdot a\right) - a}{d} \]
      mul0-lft [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \frac{4 \cdot \color{blue}{0} - a}{d} \]
      metadata-eval [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \frac{\color{blue}{0} - a}{d} \]
      neg-sub0 [<=] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \frac{\color{blue}{-a}}{d} \]
      distribute-frac-neg [=>] 9.1	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      sub-neg [<=] 9.1	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      associate-*l/ [<=] 10.1	\[ \color{blue}{\frac{c}{{d}^{2}} \cdot b} - \frac{a}{d} \]
      *-commutative [=>] 10.1	\[ \color{blue}{b \cdot \frac{c}{{d}^{2}}} - \frac{a}{d} \]
      unpow2 [=>] 10.1	\[ b \cdot \frac{c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      associate-/r* [=>] 6.1	\[ b \cdot \color{blue}{\frac{\frac{c}{d}}{d}} - \frac{a}{d} \]

   if 6.40000000000000038e-121 < c < 6.3999999999999997e69

   1. Initial program 15.7

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

   2. Applied egg-rr 11.0

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

   3. Taylor expanded in b around 0 15.7

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

   4. Simplified 12.3

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

      [Start] 15.7	\[ \frac{c \cdot b}{{d}^{2} + {c}^{2}} + -1 \cdot \frac{a \cdot d}{{d}^{2} + {c}^{2}} \]
      +-commutative [=>] 15.7	\[ \color{blue}{-1 \cdot \frac{a \cdot d}{{d}^{2} + {c}^{2}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}}} \]
      associate-/l* [=>] 13.7	\[ -1 \cdot \color{blue}{\frac{a}{\frac{{d}^{2} + {c}^{2}}{d}}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      associate-*r/ [=>] 13.7	\[ \color{blue}{\frac{-1 \cdot a}{\frac{{d}^{2} + {c}^{2}}{d}}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      mul-1-neg [=>] 13.7	\[ \frac{\color{blue}{-a}}{\frac{{d}^{2} + {c}^{2}}{d}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      +-commutative [=>] 13.7	\[ \frac{-a}{\frac{\color{blue}{{c}^{2} + {d}^{2}}}{d}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      unpow2 [=>] 13.7	\[ \frac{-a}{\frac{\color{blue}{c \cdot c} + {d}^{2}}{d}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      unpow2 [=>] 13.7	\[ \frac{-a}{\frac{c \cdot c + \color{blue}{d \cdot d}}{d}} + \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      associate-/l* [=>] 14.1	\[ \frac{-a}{\frac{c \cdot c + d \cdot d}{d}} + \color{blue}{\frac{c}{\frac{{d}^{2} + {c}^{2}}{b}}} \]
      associate-/r/ [=>] 12.3	\[ \frac{-a}{\frac{c \cdot c + d \cdot d}{d}} + \color{blue}{\frac{c}{{d}^{2} + {c}^{2}} \cdot b} \]
      +-commutative [=>] 12.3	\[ \frac{-a}{\frac{c \cdot c + d \cdot d}{d}} + \frac{c}{\color{blue}{{c}^{2} + {d}^{2}}} \cdot b \]
      unpow2 [=>] 12.3	\[ \frac{-a}{\frac{c \cdot c + d \cdot d}{d}} + \frac{c}{\color{blue}{c \cdot c} + {d}^{2}} \cdot b \]
      unpow2 [=>] 12.3	\[ \frac{-a}{\frac{c \cdot c + d \cdot d}{d}} + \frac{c}{c \cdot c + \color{blue}{d \cdot d}} \cdot b \]

   if 6.3999999999999997e69 < c

   1. Initial program 38.8

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

   2. Taylor expanded in b around inf 41.4

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

   3. Simplified 41.4

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

      [Start] 41.4	\[ \frac{c \cdot b}{{d}^{2} + {c}^{2}} \]
      unpow2 [=>] 41.4	\[ \frac{c \cdot b}{\color{blue}{d \cdot d} + {c}^{2}} \]
      unpow2 [=>] 41.4	\[ \frac{c \cdot b}{d \cdot d + \color{blue}{c \cdot c}} \]

   4. Applied egg-rr 13.8

      \[\leadsto \color{blue}{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{-121}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	13904

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := b \cdot \frac{c}{t_0} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{if}\;c \leq -6.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -1.82 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	7300

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := b \cdot \frac{c}{t_0} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{if}\;c \leq -6 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;c \leq -1.36 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 3
Error	10.1
Cost	2000

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := b \cdot \frac{c}{t_0} - \frac{a}{\frac{t_0}{d}}\\ \mathbf{if}\;c \leq -1.05 \cdot 10^{+106}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	1872

\[\begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -4.5 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{t_0}\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{d \cdot a + \left(c \cdot b - a \cdot \left(d + d\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ t_1 := c \cdot c + d \cdot d\\ t_2 := \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -9.8 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-46}:\\ \;\;\;\;b \cdot \frac{c}{t_1}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-100}:\\ \;\;\;\;\frac{a \cdot \left(-d\right)}{t_1}\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 5200:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	11.8
Cost	1488

\[\begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -4.3 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.95 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-92}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.0
Cost	1234

\[\begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{-8} \lor \neg \left(c \leq -2.7 \cdot 10^{-44}\right) \land \left(c \leq -5.9 \cdot 10^{-102} \lor \neg \left(c \leq 5.2 \cdot 10^{-49}\right)\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 8
Error	15.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ t_1 := \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -0.000175:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-20}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 53000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	15.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-47} \lor \neg \left(d \leq 1.8 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 10
Error	15.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{\frac{d}{\frac{b}{d}}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 11
Error	19.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-46} \lor \neg \left(d \leq 4.7 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 12
Error	22.8
Cost	521

\[\begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{-47} \lor \neg \left(d \leq 5.5 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13
Error	37.3
Cost	192

\[\frac{b}{c} \]

Reproduce

herbie shell --seed 2022354 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))