Complex division, real part

Average Error: 26.1 → 10.3

Time: 13.6s

Precision: binary64

Cost: 22088

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

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 -4e+277)
     (/ a (+ c (* d (/ d c))))
     (if (<= t_0 2e+282)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (+ (/ a c) (* (/ d c) (/ b 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 <= -4e+277) {
		tmp = a / (c + (d * (d / c)));
	} else if (t_0 <= 2e+282) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a / c) + ((d / c) * (b / 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 <= -4e+277)
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	elseif (t_0 <= 2e+282)
		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(d / c) * Float64(b / 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, -4e+277], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], 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[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	26.1
Target	0.4
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -4.00000000000000001e277

   1. Initial program 52.7

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

   2. Applied egg-rr 58.7

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

   3. Simplified 58.7

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

      [Start] 58.7	\[ \frac{\frac{{\left(a \cdot c\right)}^{2}}{a \cdot c - b \cdot d} - \frac{{\left(b \cdot d\right)}^{2}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      div-sub [<=] 58.7	\[ \frac{\color{blue}{\frac{{\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [<=] 58.7	\[ \frac{\frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      associate-*r/ [<=] 58.7	\[ \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [<=] 58.7	\[ \frac{\color{blue}{\left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1\right)} \cdot \frac{1}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      *-commutative [=>] 58.7	\[ \frac{\color{blue}{\left(1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)\right)} \cdot \frac{1}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      associate-*l* [=>] 58.7	\[ \frac{\color{blue}{1 \cdot \left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}\right)}}{c \cdot c + d \cdot d} \]
      metadata-eval [<=] 58.7	\[ \frac{\color{blue}{\frac{-1}{-1}} \cdot \left(\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot \frac{1}{a \cdot c - b \cdot d}\right)}{c \cdot c + d \cdot d} \]
      associate-*r/ [=>] 58.7	\[ \frac{\frac{-1}{-1} \cdot \color{blue}{\frac{\left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right) \cdot 1}{a \cdot c - b \cdot d}}}{c \cdot c + d \cdot d} \]
      *-rgt-identity [=>] 58.7	\[ \frac{\frac{-1}{-1} \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}}}{a \cdot c - b \cdot d}}{c \cdot c + d \cdot d} \]
      times-frac [<=] 58.7	\[ \frac{\color{blue}{\frac{-1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)}{-1 \cdot \left(a \cdot c - b \cdot d\right)}}}{c \cdot c + d \cdot d} \]
      neg-mul-1 [<=] 58.7	\[ \frac{\frac{-1 \cdot \left({\left(a \cdot c\right)}^{2} - {\left(b \cdot d\right)}^{2}\right)}{\color{blue}{-\left(a \cdot c - b \cdot d\right)}}}{c \cdot c + d \cdot d} \]

   4. Taylor expanded in b around 0 58.3

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

   5. Simplified 41.0

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

      [Start] 58.3	\[ \frac{c \cdot a}{{d}^{2} + {c}^{2}} \]
      *-commutative [=>] 58.3	\[ \frac{\color{blue}{a \cdot c}}{{d}^{2} + {c}^{2}} \]
      associate-/l* [=>] 41.0	\[ \color{blue}{\frac{a}{\frac{{d}^{2} + {c}^{2}}{c}}} \]
      +-commutative [=>] 41.0	\[ \frac{a}{\frac{\color{blue}{{c}^{2} + {d}^{2}}}{c}} \]
      unpow2 [=>] 41.0	\[ \frac{a}{\frac{\color{blue}{c \cdot c} + {d}^{2}}{c}} \]
      unpow2 [=>] 41.0	\[ \frac{a}{\frac{c \cdot c + \color{blue}{d \cdot d}}{c}} \]

   6. Taylor expanded in c around 0 35.9

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

   7. Simplified 35.9

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

      [Start] 35.9	\[ \frac{a}{c + \frac{{d}^{2}}{c}} \]
      unpow2 [=>] 35.9	\[ \frac{a}{c + \frac{\color{blue}{d \cdot d}}{c}} \]

   8. Applied egg-rr 32.2

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

   if -4.00000000000000001e277 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.00000000000000007e282

   1. Initial program 11.6

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

   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 2.00000000000000007e282 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 62.3

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

   2. Taylor expanded in c around inf 40.2

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

   3. Simplified 33.0

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

      [Start] 40.2	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [<=] 40.2	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      unpow2 [=>] 40.2	\[ \frac{a}{c} + \frac{b \cdot d}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 33.0	\[ \frac{a}{c} + \color{blue}{\frac{b}{c} \cdot \frac{d}{c}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.3
Cost	7300

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+71}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-91}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \end{array} \]

Alternative 2
Error	12.6
Cost	1752

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	16.5
Cost	1496

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ t_1 := \frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ t_2 := \frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 9.6 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ t_1 := \frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ t_2 := \frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{if}\;c \leq -6 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{+40}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	17.0
Cost	1496

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ t_1 := \frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{if}\;c \leq -1.56 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{d \cdot d}\\ \mathbf{elif}\;c \leq -1.32 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.0
Cost	1364

\[\begin{array}{l} t_0 := \frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{\frac{c \cdot c + d \cdot d}{b}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.3
Cost	1104

\[\begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8
Error	24.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+46}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Error	20.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;c \leq -1.18 \cdot 10^{-120} \lor \neg \left(c \leq 2.5 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Error	22.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11
Error	37.8
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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))))