Complex division, real part

Percentage Accurate: 61.7% → 86.3%

Time: 14.0s

Precision: binary64

Cost: 21961

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

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 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+303)))
     (/ (fma (/ d c) b a) c)
     (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d)))))

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)) || !(t_0 <= 2e+303)) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	}
	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)) || !(t_0 <= 2e+303))
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	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[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+303]], $MachinePrecision]], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]

Target

Original	61.7%
Target	99.3%
Herbie	86.3%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0 or 2e303 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 21.7%

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

   2. Applied egg-rr 30.5%

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

      [Start] 21.7%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 21.7%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 21.7%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 21.7%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 21.7%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 21.7%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 30.5%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Taylor expanded in c around -inf 30.0%

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

   4. Taylor expanded in c around -inf 61.8%

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

   5. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\right)} - 1} \]
      Step-by-step derivation

      [Start] 61.8%	\[ \frac{-1}{c} \cdot \left(-1 \cdot \frac{d \cdot b}{c} + -1 \cdot a\right) \]
      expm1-log1p-u [=>] 32.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{c} \cdot \left(-1 \cdot \frac{d \cdot b}{c} + -1 \cdot a\right)\right)\right)} \]
      expm1-udef [=>] 20.4%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{c} \cdot \left(-1 \cdot \frac{d \cdot b}{c} + -1 \cdot a\right)\right)} - 1} \]

   6. Simplified 70.0%

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

      [Start] 21.7%	\[ e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\right)} - 1 \]
      expm1-def [=>] 38.6%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\right)\right)} \]
      expm1-log1p [=>] 70.0%	\[ \color{blue}{\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}} \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2e303

   1. Initial program 79.5%

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

   2. Applied egg-rr 97.9%

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

      [Start] 79.5%	\[ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
      *-un-lft-identity [=>] 79.5%	\[ \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]
      add-sqr-sqrt [=>] 79.5%	\[ \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]
      times-frac [=>] 79.5%	\[ \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]
      hypot-def [=>] 79.5%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]
      fma-def [=>] 79.5%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]
      hypot-def [=>] 97.9%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 98.0%

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

      [Start] 97.9%	\[ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]
      associate-*l/ [=>] 98.0%	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}} \]
      *-un-lft-identity [<=] 98.0%	\[ \frac{\color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.3%
Cost	21961

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2
Accuracy	82.4%
Cost	7568

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq -1.12 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-148}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3
Accuracy	82.7%
Cost	7568

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-a\right) - d \cdot \frac{b}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4
Accuracy	82.0%
Cost	7376

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{if}\;c \leq -7.6 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	82.0%
Cost	7376

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq -2.25 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-144}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]

Alternative 6
Accuracy	82.3%
Cost	1488

\[\begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{1}{c}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.02 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	67.8%
Cost	1100

\[\begin{array}{l} t_0 := \left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{1}{c}\\ \mathbf{if}\;c \leq -6 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-175}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{a \cdot c}{d} \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	61.1%
Cost	972

\[\begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{a \cdot c}{d} \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Accuracy	76.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{-30} \lor \neg \left(c \leq 3.7 \cdot 10^{-65}\right):\\ \;\;\;\;\left(a + \frac{d}{\frac{c}{b}}\right) \cdot \frac{1}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{\frac{c}{d}}{d}\\ \end{array} \]

Alternative 10
Accuracy	62.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{+127}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11
Accuracy	43.3%
Cost	192

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

Reproduce

herbie shell --seed 2023263 
(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))))