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

Average Error: 0.0 → 0.0

Time: 3.9s

Precision: binary64

Cost: 6976

Math

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

↓

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

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(x \cdot y + z \cdot t\right) + a \cdot b \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} + a \cdot b \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-112} \lor \neg \left(t \leq 4.9 \cdot 10^{+154}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	36.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-96}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 4
Error	21.2
Cost	448

\[a \cdot b + z \cdot t \]

Alternative 5
Error	42.1
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2022354 
(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)))