Complex division, real part

Average Error: 26.0 → 9.6

Time: 11.2s

Precision: binary64

Cost: 22088

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;t_0 \leq 10^{+287}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

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 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 (- INFINITY))
     (+ (/ b d) (* (/ a d) (/ c d)))
     (if (<= t_0 1e+287)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (+ (/ a c) (* (/ b c) (/ d c)))))))

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 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (b / d) + ((a / d) * (c / d));
	} else if (t_0 <= 1e+287) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	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(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(b / d) + Float64(Float64(a / d) * Float64(c / d)));
	elseif (t_0 <= 1e+287)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	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), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(b / d), $MachinePrecision] + N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+287], 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], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	26.0
Target	0.5
Herbie	9.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in c around 0 46.3

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

   3. Simplified 35.5

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

      [Start] 46.3	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      *-commutative [<=] 46.3	\[ \frac{b}{d} + \frac{\color{blue}{a \cdot c}}{{d}^{2}} \]
      unpow2 [=>] 46.3	\[ \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 35.5	\[ \frac{b}{d} + \color{blue}{\frac{a}{d} \cdot \frac{c}{d}} \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.0000000000000001e287

   1. Initial program 11.5

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

   2. Applied egg-rr 0.9

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

   if 1.0000000000000001e287 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 63.1

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

   2. Taylor expanded in c around inf 38.8

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

   3. Simplified 31.6

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

      [Start] 38.8	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [<=] 38.8	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      unpow2 [=>] 38.8	\[ \frac{a}{c} + \frac{b \cdot d}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 31.6	\[ \frac{a}{c} + \color{blue}{\frac{b}{c} \cdot \frac{d}{c}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -\infty:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+287}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	1356

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;c \leq -300000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-225}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	18.9
Cost	1232

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;d \leq -4.3 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 3
Error	21.4
Cost	1100

\[\begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+117}:\\ \;\;\;\;a \cdot \frac{c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 4
Error	19.0
Cost	1100

\[\begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+43}:\\ \;\;\;\;a \cdot \frac{c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -300000000000 \lor \neg \left(c \leq 8.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \end{array} \]

Alternative 6
Error	15.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -270000000000 \lor \neg \left(c \leq 1.2 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \end{array} \]

Alternative 7
Error	36.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+207}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+215}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 8
Error	22.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2023041 
(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))))