Complex division, real part

Average Error: 26.6 → 15.3

Time: 11.3s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{if}\;d \leq -64806.10812830801:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.1912272487295739 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}}\\ \mathbf{elif}\;d \leq -4.161477497812156 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ \mathbf{elif}\;d \leq 2.864392280568695 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, d \cdot \frac{d}{c}, c\right)} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (+ (/ b d) (/ c (/ d (/ a d))))))
   (if (<= d -64806.10812830801)
     t_0
     (if (<= d -1.1912272487295739e-50)
       (/ (/ d (hypot d c)) (/ (hypot d c) b))
       (if (<= d -4.161477497812156e-67)
         (+ (/ b d) (/ (/ a d) (/ d c)))
         (if (<= d 2.864392280568695e-37)
           (* (/ 1.0 (fma 0.5 (* d (/ d c)) c)) (+ a (/ (* d b) c)))
           t_0))))))

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 = (b / d) + (c / (d / (a / d)));
	double tmp;
	if (d <= -64806.10812830801) {
		tmp = t_0;
	} else if (d <= -1.1912272487295739e-50) {
		tmp = (d / hypot(d, c)) / (hypot(d, c) / b);
	} else if (d <= -4.161477497812156e-67) {
		tmp = (b / d) + ((a / d) / (d / c));
	} else if (d <= 2.864392280568695e-37) {
		tmp = (1.0 / fma(0.5, (d * (d / c)), c)) * (a + ((d * b) / c));
	} else {
		tmp = t_0;
	}
	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(b / d) + Float64(c / Float64(d / Float64(a / d))))
	tmp = 0.0
	if (d <= -64806.10812830801)
		tmp = t_0;
	elseif (d <= -1.1912272487295739e-50)
		tmp = Float64(Float64(d / hypot(d, c)) / Float64(hypot(d, c) / b));
	elseif (d <= -4.161477497812156e-67)
		tmp = Float64(Float64(b / d) + Float64(Float64(a / d) / Float64(d / c)));
	elseif (d <= 2.864392280568695e-37)
		tmp = Float64(Float64(1.0 / fma(0.5, Float64(d * Float64(d / c)), c)) * Float64(a + Float64(Float64(d * b) / c)));
	else
		tmp = t_0;
	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[(b / d), $MachinePrecision] + N[(c / N[(d / N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -64806.10812830801], t$95$0, If[LessEqual[d, -1.1912272487295739e-50], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.161477497812156e-67], N[(N[(b / d), $MachinePrecision] + N[(N[(a / d), $MachinePrecision] / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.864392280568695e-37], N[(N[(1.0 / N[(0.5 * N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] * N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Target

Original	26.6
Target	0.3
Herbie	15.3

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

Derivation

1. Split input into 4 regimes

2. if d < -64806.108128308013 or 2.864392280568695e-37 < d

   1. Initial program 31.9

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

   2. Applied egg-rr 21.8

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

   3. Taylor expanded in c around 0 20.5

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

   4. Simplified 16.4

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

   5. Applied egg-rr 17.2

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

   if -64806.108128308013 < d < -1.1912272487295739e-50

   1. Initial program 13.4

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

   2. Taylor expanded in a around 0 29.5

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

   3. Applied egg-rr 27.5

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

   4. Applied egg-rr 27.6

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

   if -1.1912272487295739e-50 < d < -4.1614774978121557e-67

   1. Initial program 14.4

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

   2. Applied egg-rr 12.0

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

   3. Taylor expanded in c around 0 37.9

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

   4. Simplified 39.6

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

   5. Applied egg-rr 39.6

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

   if -4.1614774978121557e-67 < d < 2.864392280568695e-37

   1. Initial program 20.5

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

   2. Applied egg-rr 12.1

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

   3. Taylor expanded in c around inf 32.8

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

   4. Taylor expanded in c around inf 10.5

      \[\leadsto \frac{1}{\color{blue}{c + 0.5 \cdot \frac{{d}^{2}}{c}}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]

   5. Simplified 10.2

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{d}{c} \cdot d, c\right)}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -64806.10812830801:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d}{\frac{a}{d}}}\\ \mathbf{elif}\;d \leq -1.1912272487295739 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{b}}\\ \mathbf{elif}\;d \leq -4.161477497812156 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ \mathbf{elif}\;d \leq 2.864392280568695 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, d \cdot \frac{d}{c}, c\right)} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d}{\frac{a}{d}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	13772

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ t_1 := \frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{if}\;c \leq -1.5486162962968479 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.936106704635713 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.1193682608307834 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 1.413535944840097 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	22.2
Cost	1364

\[\begin{array}{l} t_0 := \frac{c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.5486162962968479 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -5.183884978967185 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq -1.412094686123106 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.602876509221542 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.508885696595655 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 3
Error	19.5
Cost	1232

\[\begin{array}{l} t_0 := \left(a + \frac{d \cdot b}{c}\right) \cdot \frac{1}{c}\\ \mathbf{if}\;d \leq -7.90595019832564 \cdot 10^{+49}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3.1563451937418893 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.816835379553297 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.864392280568695 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 4
Error	19.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{if}\;d \leq -7.90595019832564 \cdot 10^{+49}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3.1563451937418893 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.816835379553297 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.864392280568695 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5
Error	16.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ t_1 := \frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{if}\;c \leq -1.5486162962968479 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.936106704635713 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.1193682608307834 \cdot 10^{+67}:\\ \;\;\;\;\frac{c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.413535944840097 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	22.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -3.0397468271869814 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.864392280568695 \cdot 10^{-37}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7
Error	37.0
Cost	192

\[\frac{b}{d} \]

Reproduce

herbie shell --seed 2022228 
(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))))