Complex division, real part

Percentage Accurate: 62.6% → 84.9%

Time: 16.2s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := t_0 \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -6 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+174}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

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)))
        (t_1 (* t_0 (/ (fma a c (* d b)) (hypot c d)))))
   (if (<= d -6e+74)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -4.4e-102)
       t_1
       (if (<= d 2.2e-203)
         (+ (/ a c) (/ (* d (/ b c)) c))
         (if (<= d 1.45e+174) t_1 (* t_0 (+ b (/ c (/ 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);
	double t_1 = t_0 * (fma(a, c, (d * b)) / hypot(c, d));
	double tmp;
	if (d <= -6e+74) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -4.4e-102) {
		tmp = t_1;
	} else if (d <= 2.2e-203) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= 1.45e+174) {
		tmp = t_1;
	} else {
		tmp = t_0 * (b + (c / (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(1.0 / hypot(c, d))
	t_1 = Float64(t_0 * Float64(fma(a, c, Float64(d * b)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -6e+74)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -4.4e-102)
		tmp = t_1;
	elseif (d <= 2.2e-203)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (d <= 1.45e+174)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(b + Float64(c / Float64(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[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6e+74], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.4e-102], t$95$1, If[LessEqual[d, 2.2e-203], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.45e+174], t$95$1, N[(t$95$0 * N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	62.6%
Target	99.3%
Herbie	84.9%

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

Derivation

1. Split input into 4 regimes

2. if d < -6e74

   1. Initial program 47.5%

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

   2. Taylor expanded in c around 0 84.0%

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

   3. Simplified 88.6%

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

      [Start] 84.0	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      +-commutative [=>] 84.0	\[ \color{blue}{\frac{c \cdot a}{{d}^{2}} + \frac{b}{d}} \]
      unpow2 [=>] 84.0	\[ \frac{c \cdot a}{\color{blue}{d \cdot d}} + \frac{b}{d} \]
      times-frac [=>] 88.6	\[ \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} + \frac{b}{d} \]
      fma-def [=>] 88.6	\[ \color{blue}{\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)} \]

   4. Applied egg-rr 88.6%

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

      [Start] 88.6	\[ \mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right) \]
      fma-udef [=>] 88.6	\[ \color{blue}{\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}} \]
      +-commutative [=>] 88.6	\[ \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]

   if -6e74 < d < -4.40000000000000026e-102 or 2.2e-203 < d < 1.45e174

   1. Initial program 80.2%

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

   2. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]
      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 [=>] 80.2	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 80.3	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 80.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 91.4	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -4.40000000000000026e-102 < d < 2.2e-203

   1. Initial program 66.4%

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

   2. Taylor expanded in c around inf 75.2%

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

   3. Simplified 75.2%

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

      [Start] 75.2	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      unpow2 [=>] 75.2	\[ \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

   4. Taylor expanded in d around 0 75.2%

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

   5. Simplified 75.4%

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

      [Start] 75.2	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      unpow2 [=>] 75.2	\[ \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]
      associate-*r/ [<=] 75.4	\[ \frac{a}{c} + \color{blue}{d \cdot \frac{b}{c \cdot c}} \]

   6. Applied egg-rr 90.9%

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

      [Start] 75.4	\[ \frac{a}{c} + d \cdot \frac{b}{c \cdot c} \]
      *-commutative [=>] 75.4	\[ \frac{a}{c} + \color{blue}{\frac{b}{c \cdot c} \cdot d} \]
      associate-/r* [=>] 84.4	\[ \frac{a}{c} + \color{blue}{\frac{\frac{b}{c}}{c}} \cdot d \]
      associate-*l/ [=>] 90.9	\[ \frac{a}{c} + \color{blue}{\frac{\frac{b}{c} \cdot d}{c}} \]

   if 1.45e174 < d

   1. Initial program 33.1%

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

   2. Applied egg-rr 56.6%

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

      [Start] 33.1	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 33.1	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 33.1	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 33.1	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 33.1	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 33.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 56.6	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around 0 76.2%

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

   4. Simplified 93.1%

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

      [Start] 76.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
      associate-/l* [=>] 93.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \color{blue}{\frac{c}{\frac{d}{a}}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.0%
Cost	2000

\[\begin{array}{l} t_0 := d \cdot d + c \cdot c\\ t_1 := \frac{d}{\frac{t_0}{b}} + \frac{c}{\frac{t_0}{a}}\\ t_2 := \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -9 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	81.6%
Cost	1488

\[\begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -7.3 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	74.9%
Cost	1233

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.52 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-87} \lor \neg \left(d \leq 3.8 \cdot 10^{+26}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 4
Accuracy	75.1%
Cost	1232

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-85}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	75.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	67.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{-21} \lor \neg \left(c \leq 2.05 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Accuracy	69.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{-51} \lor \neg \left(d \leq 3 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 8
Accuracy	70.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{-51} \lor \neg \left(d \leq 1.05 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 9
Accuracy	62.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-82}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 24000000000:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Accuracy	42.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 11
Accuracy	42.4%
Cost	192

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

Reproduce

herbie shell --seed 2023160 
(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))))