Complex division, real part

Average Accuracy: 58.0% → 83.9%

Time: 13.1s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -5.7 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-126}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* b d)) (hypot c d))))
        (t_1 (+ (/ a c) (* (/ b c) (/ d c)))))
   (if (<= c -5.7e+181)
     t_1
     (if (<= c -3.2e-147)
       t_0
       (if (<= c 1e-126)
         (+ (/ b d) (* a (/ (/ c d) d)))
         (if (<= c 5e+44) t_0 t_1))))))

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, (b * d)) / hypot(c, d));
	double t_1 = (a / c) + ((b / c) * (d / c));
	double tmp;
	if (c <= -5.7e+181) {
		tmp = t_1;
	} else if (c <= -3.2e-147) {
		tmp = t_0;
	} else if (c <= 1e-126) {
		tmp = (b / d) + (a * ((c / d) / d));
	} else if (c <= 5e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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(b * d)) / hypot(c, d)))
	t_1 = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -5.7e+181)
		tmp = t_1;
	elseif (c <= -3.2e-147)
		tmp = t_0;
	elseif (c <= 1e-126)
		tmp = Float64(Float64(b / d) + Float64(a * Float64(Float64(c / d) / d)));
	elseif (c <= 5e+44)
		tmp = t_0;
	else
		tmp = t_1;
	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[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.7e+181], t$95$1, If[LessEqual[c, -3.2e-147], t$95$0, If[LessEqual[c, 1e-126], N[(N[(b / d), $MachinePrecision] + N[(a * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5e+44], t$95$0, t$95$1]]]]]]

Target

Original	58.0%
Target	99.4%
Herbie	83.9%

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

Derivation

1. Split input into 3 regimes

2. if c < -5.7000000000000002e181 or 4.9999999999999996e44 < c

   1. Initial program 40.0%

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

   2. Taylor expanded in c around inf 74.9%

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

   3. Simplified 83.9%

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

      [Start] 74.9	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [<=] 74.9	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      unpow2 [=>] 74.9	\[ \frac{a}{c} + \frac{b \cdot d}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 83.9	\[ \frac{a}{c} + \color{blue}{\frac{b}{c} \cdot \frac{d}{c}} \]

   if -5.7000000000000002e181 < c < -3.19999999999999979e-147 or 9.9999999999999995e-127 < c < 4.9999999999999996e44

   1. Initial program 70.4%

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

   2. Applied egg-rr 80.5%

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

      [Start] 70.4	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 70.4	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 70.4	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 70.3	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 70.3	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 70.3	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 80.5	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   if -3.19999999999999979e-147 < c < 9.9999999999999995e-127

   1. Initial program 62.2%

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

   2. Applied egg-rr 79.1%

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

      [Start] 62.2	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 62.2	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 62.2	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 62.2	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 62.2	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 62.2	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 79.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 79.0%

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

      [Start] 79.1	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]
      clear-num [=>] 79.0	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}} \]
      un-div-inv [=>] 79.0	\[ \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}} \]

   4. Taylor expanded in c around 0 84.7%

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

   5. Simplified 89.5%

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

      [Start] 84.7	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      associate-/l* [=>] 79.5	\[ \frac{b}{d} + \color{blue}{\frac{c}{\frac{{d}^{2}}{a}}} \]
      associate-/r/ [=>] 83.4	\[ \frac{b}{d} + \color{blue}{\frac{c}{{d}^{2}} \cdot a} \]
      unpow2 [=>] 83.4	\[ \frac{b}{d} + \frac{c}{\color{blue}{d \cdot d}} \cdot a \]
      associate-/r* [=>] 89.5	\[ \frac{b}{d} + \color{blue}{\frac{\frac{c}{d}}{d}} \cdot a \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 10^{-126}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.4%
Cost	1488

\[\begin{array}{l} t_0 := \frac{b \cdot d + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 2
Accuracy	75.9%
Cost	1232

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -41000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	66.2%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;c \leq -37000:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+120}:\\ \;\;\;\;a \cdot \frac{c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 4
Accuracy	71.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -4.8 \lor \neg \left(c \leq 6.3 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5
Accuracy	71.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -0.122 \lor \neg \left(c \leq 7.2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 6
Accuracy	76.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{+18} \lor \neg \left(c \leq 6.6 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 7
Accuracy	65.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -200:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8
Accuracy	41.9%
Cost	192

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

Reproduce

herbie shell --seed 2023126 
(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))))