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

Percentage Accurate: 97.6% → 98.6%

Time: 6.7s

Precision: binary64

Cost: 13248

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

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 (fma x y (fma z t (* 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) {
	return fma(x, y, fma(z, t, (a * b)));
}

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)
	return fma(x, y, fma(z, t, Float64(a * b)))
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_] := N[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

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

2. Simplified 99.2%

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

   [Start] 97.2%	\[ \left(x \cdot y + z \cdot t\right) + a \cdot b \]
   associate-+l+ [=>] 97.2%	\[ \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]
   fma-def [=>] 98.4%	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + a \cdot b\right)} \]
   fma-def [=>] 99.2%	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

3. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	98.6%
Cost	13248

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

Alternative 2
Accuracy	98.6%
Cost	7492

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

Alternative 3
Accuracy	97.9%
Cost	6976

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

Alternative 4
Accuracy	98.0%
Cost	6976

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

Alternative 5
Accuracy	83.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.8 \cdot 10^{+103}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.55 \cdot 10^{-83}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 6
Accuracy	78.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+127} \lor \neg \left(z \leq 4.1 \cdot 10^{-19}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 7
Accuracy	53.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.08 \cdot 10^{+52}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8
Accuracy	71.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+122}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-11}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Accuracy	97.6%
Cost	704

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

Alternative 10
Accuracy	48.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2500000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+39}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Accuracy	35.0%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023271 
(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)))