Complex division, imag part

Percentage Accurate: 61.5% → 95.0%

Time: 16.8s

Precision: binary64

Cost: 46476

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-251}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{\sqrt{d} \cdot \frac{-a}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right)\\ \end{array} \]

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 (hypot c d))) (t_1 (/ b (hypot c d))))
   (if (<= d -1.3e+154)
     (- (* (/ c d) (/ b d)) (/ a d))
     (if (<= d -1e-180)
       (fma t_0 t_1 (/ (- a) (/ (pow (hypot c d) 2.0) d)))
       (if (<= d 2e-251)
         (/ (- b (/ a (/ c d))) c)
         (fma
          t_0
          t_1
          (/ (* (sqrt d) (/ (- a) (hypot c d))) (/ (hypot c d) (sqrt 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 / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double tmp;
	if (d <= -1.3e+154) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else if (d <= -1e-180) {
		tmp = fma(t_0, t_1, (-a / (pow(hypot(c, d), 2.0) / d)));
	} else if (d <= 2e-251) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = fma(t_0, t_1, ((sqrt(d) * (-a / hypot(c, d))) / (hypot(c, d) / sqrt(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(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	tmp = 0.0
	if (d <= -1.3e+154)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	elseif (d <= -1e-180)
		tmp = fma(t_0, t_1, Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)));
	elseif (d <= 2e-251)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = fma(t_0, t_1, Float64(Float64(sqrt(d) * Float64(Float64(-a) / hypot(c, d))) / Float64(hypot(c, d) / sqrt(d))));
	end
	return 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[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e+154], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-180], N[(t$95$0 * t$95$1 + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e-251], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(t$95$0 * t$95$1 + N[(N[(N[Sqrt[d], $MachinePrecision] * N[((-a) / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	61.5%
Target	99.4%
Herbie	95.0%

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

Derivation

1. Split input into 4 regimes

2. if d < -1.29999999999999994e154

   1. Initial program 23.3%

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

   2. Taylor expanded in c around 0 72.6%

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

   3. Simplified 83.5%

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

      [Start] 72.6%	\[ -1 \cdot \frac{a}{d} + \frac{c \cdot b}{{d}^{2}} \]
      +-commutative [=>] 72.6%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]
      mul-1-neg [=>] 72.6%	\[ \frac{c \cdot b}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]
      unsub-neg [=>] 72.6%	\[ \color{blue}{\frac{c \cdot b}{{d}^{2}} - \frac{a}{d}} \]
      unpow2 [=>] 72.6%	\[ \frac{c \cdot b}{\color{blue}{d \cdot d}} - \frac{a}{d} \]
      times-frac [=>] 83.5%	\[ \color{blue}{\frac{c}{d} \cdot \frac{b}{d}} - \frac{a}{d} \]

   if -1.29999999999999994e154 < d < -1e-180

   1. Initial program 79.3%

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

   2. Applied egg-rr 93.1%

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

      [Start] 79.3%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      div-sub [=>] 77.8%	\[ \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}} \]
      *-commutative [=>] 77.8%	\[ \frac{\color{blue}{c \cdot b}}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 77.8%	\[ \frac{c \cdot b}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      times-frac [=>] 78.4%	\[ \color{blue}{\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      fma-neg [=>] 78.4%	\[ \color{blue}{\mathsf{fma}\left(\frac{c}{\sqrt{c \cdot c + d \cdot d}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right)} \]
      hypot-def [=>] 78.4%	\[ \mathsf{fma}\left(\frac{c}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      hypot-def [=>] 89.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      associate-/l* [=>] 93.1%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\right) \]
      add-sqr-sqrt [=>] 93.1%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}{d}}\right) \]
      pow2 [=>] 93.1%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{{\left(\sqrt{c \cdot c + d \cdot d}\right)}^{2}}}{d}}\right) \]
      hypot-def [=>] 93.1%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}}{d}}\right) \]

   if -1e-180 < d < 2.00000000000000003e-251

   1. Initial program 65.9%

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

   2. Applied egg-rr 78.2%

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

      [Start] 65.9%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      div-sub [=>] 60.2%	\[ \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}} \]
      *-commutative [=>] 60.2%	\[ \frac{\color{blue}{c \cdot b}}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 60.2%	\[ \frac{c \cdot b}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      times-frac [=>] 68.2%	\[ \color{blue}{\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      fma-neg [=>] 68.2%	\[ \color{blue}{\mathsf{fma}\left(\frac{c}{\sqrt{c \cdot c + d \cdot d}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right)} \]
      hypot-def [=>] 68.2%	\[ \mathsf{fma}\left(\frac{c}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      hypot-def [=>] 80.5%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      associate-/l* [=>] 78.2%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\right) \]
      add-sqr-sqrt [=>] 78.2%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}{d}}\right) \]
      pow2 [=>] 78.2%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{{\left(\sqrt{c \cdot c + d \cdot d}\right)}^{2}}}{d}}\right) \]
      hypot-def [=>] 78.2%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}}{d}}\right) \]

   3. Taylor expanded in c around inf 80.5%

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

   4. Simplified 80.5%

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

      [Start] 80.5%	\[ -1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c} \]
      +-commutative [=>] 80.5%	\[ \color{blue}{\frac{b}{c} + -1 \cdot \frac{a \cdot d}{{c}^{2}}} \]
      mul-1-neg [=>] 80.5%	\[ \frac{b}{c} + \color{blue}{\left(-\frac{a \cdot d}{{c}^{2}}\right)} \]
      unsub-neg [=>] 80.5%	\[ \color{blue}{\frac{b}{c} - \frac{a \cdot d}{{c}^{2}}} \]
      *-commutative [<=] 80.5%	\[ \frac{b}{c} - \frac{\color{blue}{d \cdot a}}{{c}^{2}} \]
      unpow2 [=>] 80.5%	\[ \frac{b}{c} - \frac{d \cdot a}{\color{blue}{c \cdot c}} \]

   5. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{1 \cdot \frac{b - \frac{d \cdot a}{c}}{c}} \]
      Step-by-step derivation

      [Start] 80.5%	\[ \frac{b}{c} - \frac{d \cdot a}{c \cdot c} \]
      *-un-lft-identity [=>] 80.5%	\[ \color{blue}{1 \cdot \left(\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\right)} \]
      associate-/r* [=>] 98.3%	\[ 1 \cdot \left(\frac{b}{c} - \color{blue}{\frac{\frac{d \cdot a}{c}}{c}}\right) \]
      sub-div [=>] 98.3%	\[ 1 \cdot \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]

   6. Simplified 98.3%

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

      [Start] 98.3%	\[ 1 \cdot \frac{b - \frac{d \cdot a}{c}}{c} \]
      *-lft-identity [=>] 98.3%	\[ \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
      *-commutative [<=] 98.3%	\[ \frac{b - \frac{\color{blue}{a \cdot d}}{c}}{c} \]
      associate-/l* [=>] 98.3%	\[ \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   if 2.00000000000000003e-251 < d

   1. Initial program 58.5%

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

   2. Applied egg-rr 76.9%

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

      [Start] 58.5%	\[ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
      div-sub [=>] 56.8%	\[ \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}} \]
      *-commutative [=>] 56.8%	\[ \frac{\color{blue}{c \cdot b}}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 56.7%	\[ \frac{c \cdot b}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      times-frac [=>] 60.8%	\[ \color{blue}{\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}}} - \frac{a \cdot d}{c \cdot c + d \cdot d} \]
      fma-neg [=>] 60.8%	\[ \color{blue}{\mathsf{fma}\left(\frac{c}{\sqrt{c \cdot c + d \cdot d}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right)} \]
      hypot-def [=>] 60.8%	\[ \mathsf{fma}\left(\frac{c}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, \frac{b}{\sqrt{c \cdot c + d \cdot d}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      hypot-def [=>] 74.6%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\color{blue}{\mathsf{hypot}\left(c, d\right)}}, -\frac{a \cdot d}{c \cdot c + d \cdot d}\right) \]
      associate-/l* [=>] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\right) \]
      add-sqr-sqrt [=>] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}{d}}\right) \]
      pow2 [=>] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{\color{blue}{{\left(\sqrt{c \cdot c + d \cdot d}\right)}^{2}}}{d}}\right) \]
      hypot-def [=>] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}}{d}}\right) \]

   3. Applied egg-rr 96.7%

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

      [Start] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right) \]
      *-un-lft-identity [=>] 76.9%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{\color{blue}{1 \cdot a}}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right) \]
      add-sqr-sqrt [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1 \cdot a}{\color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}} \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}}\right) \]
      times-frac [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{1}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}}\right) \]
      unpow2 [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\sqrt{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right) \cdot \mathsf{hypot}\left(c, d\right)}}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      hypot-udef [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\sqrt{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d}} \cdot \mathsf{hypot}\left(c, d\right)}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      hypot-udef [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\sqrt{\frac{\sqrt{c \cdot c + d \cdot d} \cdot \color{blue}{\sqrt{c \cdot c + d \cdot d}}}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      add-sqr-sqrt [<=] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\sqrt{\frac{\color{blue}{c \cdot c + d \cdot d}}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      +-commutative [<=] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\sqrt{\frac{\color{blue}{d \cdot d + c \cdot c}}{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      sqrt-div [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\color{blue}{\frac{\sqrt{d \cdot d + c \cdot c}}{\sqrt{d}}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      +-commutative [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\sqrt{\color{blue}{c \cdot c + d \cdot d}}}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      hypot-udef [<=] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right)}}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}}\right) \]
      unpow2 [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right) \cdot \mathsf{hypot}\left(c, d\right)}}{d}}}\right) \]
      hypot-udef [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{\color{blue}{\sqrt{c \cdot c + d \cdot d}} \cdot \mathsf{hypot}\left(c, d\right)}{d}}}\right) \]
      hypot-udef [=>] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{\sqrt{c \cdot c + d \cdot d} \cdot \color{blue}{\sqrt{c \cdot c + d \cdot d}}}{d}}}\right) \]
      add-sqr-sqrt [<=] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{\color{blue}{c \cdot c + d \cdot d}}{d}}}\right) \]
      +-commutative [<=] 76.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\sqrt{\frac{\color{blue}{d \cdot d + c \cdot c}}{d}}}\right) \]

   4. Simplified 96.8%

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

      [Start] 96.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}} \cdot \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right) \]
      associate-*l/ [=>] 96.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{1 \cdot \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}}\right) \]
      *-lft-identity [=>] 96.7%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{\color{blue}{\frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right) \]
      associate-/r/ [=>] 96.8%	\[ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\frac{\color{blue}{\frac{a}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{d}}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-251}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{\sqrt{d} \cdot \frac{-a}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.0%
Cost	46476

\[\begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-251}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{\sqrt{d} \cdot \frac{-a}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{\sqrt{d}}}\right)\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	14660

\[\begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]

Alternative 3
Accuracy	78.9%
Cost	13900

\[\begin{array}{l} \mathbf{if}\;c \leq -1.28 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{d}{c}}{\frac{c}{a}}\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{+66}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4
Accuracy	78.8%
Cost	7436

\[\begin{array}{l} \mathbf{if}\;c \leq -6.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{d}{c}}{\frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+66}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 5
Accuracy	78.8%
Cost	1356

\[\begin{array}{l} t_0 := \frac{b}{c} - \frac{\frac{d}{c}}{\frac{c}{a}}\\ \mathbf{if}\;c \leq -4 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.36 \cdot 10^{+112}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	75.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{-33} \lor \neg \left(c \leq 1.55 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 7
Accuracy	76.0%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \leq -6.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{d}{c}}{\frac{c}{a}}\\ \mathbf{elif}\;c \leq 3.15 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]

Alternative 8
Accuracy	71.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+115} \lor \neg \left(d \leq 3.1 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]

Alternative 9
Accuracy	63.4%
Cost	520

\[\begin{array}{l} \mathbf{if}\;c \leq -66000000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 10
Accuracy	10.1%
Cost	192

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

Alternative 11
Accuracy	43.4%
Cost	192

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

Reproduce

herbie shell --seed 2023263 
(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))))