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

Average Accuracy: 100.0% → 100.0%

Time: 5.2s

Precision: binary64

Cost: 6976

Math

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

↓

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

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x \cdot y + z \cdot t\right) + a \cdot b \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} + a \cdot b \]

3. Taylor expanded in x around 0 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ \left(y \cdot x + t \cdot z\right) + a \cdot b \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(t \cdot z + y \cdot x\right)} + a \cdot b \]
   *-commutative [<=] 100.0	\[ \left(\color{blue}{z \cdot t} + y \cdot x\right) + a \cdot b \]
   *-commutative [=>] 100.0	\[ \left(z \cdot t + \color{blue}{x \cdot y}\right) + a \cdot b \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z, t, x \cdot y\right)} + a \cdot b \]
   *-commutative [<=] 100.0	\[ \mathsf{fma}\left(z, t, \color{blue}{y \cdot x}\right) + a \cdot b \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	52.2%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.5 \cdot 10^{-127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-309}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 2.1 \cdot 10^{-94}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 0.0033:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 3
Accuracy	74.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+74} \lor \neg \left(x \leq 1.3 \cdot 10^{-225}\right):\\ \;\;\;\;y \cdot x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 4
Accuracy	51.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8.4 \cdot 10^{-14}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5
Accuracy	66.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+104}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-71}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	100.0%
Cost	704

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

Alternative 7
Accuracy	35.0%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023129 
(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)))