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

Average Accuracy: 97.7% → 98.9%

Time: 4.5s

Precision: binary64

Cost: 7492

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

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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) (+ (* z t) (* x y)))))
   (if (<= t_1 INFINITY) t_1 (fma y x (* a b)))))

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) + ((z * t) + (x * y));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(y, x, (a * b));
	}
	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(z * t) + Float64(x * y)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(y, x, Float64(a * b));
	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[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(y * x + N[(a * b), $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 z around 0 30.0%

      \[\leadsto \color{blue}{a \cdot b + y \cdot x} \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, a \cdot b\right)} \]
      Proof

      [Start] 30.0	\[ a \cdot b + y \cdot x \]
      +-commutative [=>] 30.0	\[ \color{blue}{y \cdot x + a \cdot b} \]
      fma-def [=>] 60.0	\[ \color{blue}{\mathsf{fma}\left(y, x, a \cdot b\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	13248

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

Alternative 2
Accuracy	98.5%
Cost	1476

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

Alternative 3
Accuracy	53.7%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.7 \cdot 10^{-149}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3.3 \cdot 10^{+85}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 4
Accuracy	78.2%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-67} \lor \neg \left(y \leq 2.4 \cdot 10^{+50} \lor \neg \left(y \leq 1.05 \cdot 10^{+184}\right) \land y \leq 5 \cdot 10^{+207}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 5
Accuracy	85.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -7.8 \cdot 10^{+47} \lor \neg \left(a \cdot b \leq 4.5 \cdot 10^{+83}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \]

Alternative 6
Accuracy	67.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-64} \lor \neg \left(y \leq 9 \cdot 10^{+207}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 7
Accuracy	54.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7.8 \cdot 10^{+89}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8
Accuracy	34.5%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023160 
(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)))