Linear.V3:$cdot from linear-1.19.1.3, B

Percentage Accurate: 98.1% → 99.0%

Time: 7.1s

Precision: binary64

Cost: 7492

• Unsound rule application detected in e-graph. Results may not be sound.

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

↓

\[\begin{array}{l} t_1 := a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (+ (* x y) (* z t)))))
   (if (<= t_1 INFINITY) t_1 (fma t z (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((x * y) + (z * t));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(t, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(Float64(x * y) + Float64(z * t)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(t, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) < +inf.0

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   if +inf.0 < (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b))

   1. Initial program 0.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in a around 0 33.3%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, y \cdot x\right)} \]
      Step-by-step derivation

      [Start] 33.3%	\[ y \cdot x + t \cdot z \]
      +-commutative [=>] 33.3%	\[ \color{blue}{t \cdot z + y \cdot x} \]
      *-commutative [=>] 33.3%	\[ t \cdot z + \color{blue}{x \cdot y} \]
      fma-udef [<=] 66.7%	\[ \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]
      *-commutative [<=] 66.7%	\[ \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b + \left(x \cdot y + z \cdot t\right) \leq \infty:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	7492

\[\begin{array}{l} t_1 := a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	13248

\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, a \cdot b\right)\right) \]

Alternative 3
Accuracy	98.3%
Cost	6976

\[a \cdot b + \mathsf{fma}\left(x, y, z \cdot t\right) \]

Alternative 4
Accuracy	53.7%
Cost	2272

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.45 \cdot 10^{+131}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -1.02 \cdot 10^{-141}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.5 \cdot 10^{-246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 1.7 \cdot 10^{-134}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.7 \cdot 10^{-77}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 6 \cdot 10^{+54}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5
Accuracy	84.4%
Cost	1489

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.8 \cdot 10^{+77}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.02 \cdot 10^{-52} \lor \neg \left(a \cdot b \leq -1.05 \cdot 10^{-87}\right) \land a \cdot b \leq 1.15 \cdot 10^{-26}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 6
Accuracy	53.9%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+75}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.02 \cdot 10^{-52} \lor \neg \left(a \cdot b \leq -4.8 \cdot 10^{-77}\right) \land a \cdot b \leq 0.019:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7
Accuracy	77.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-60} \lor \neg \left(t \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 8
Accuracy	70.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+137}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Accuracy	98.1%
Cost	704

\[a \cdot b + \left(x \cdot y + z \cdot t\right) \]

Alternative 10
Accuracy	35.8%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023277 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))