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

Average Error: 0.03% → 0.01%

Time: 5.4s

Precision: binary64

Cost: 13248

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 0.03

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

2. Simplified 0.01

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

   [Start] 0.03	\[ \left(x \cdot y + z \cdot t\right) + a \cdot b \]
   associate-+l+ [=>] 0.03	\[ \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]
   fma-def [=>] 0.02	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + a \cdot b\right)} \]
   fma-def [=>] 0.01	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	49.11%
Cost	3053

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.2 \cdot 10^{+75}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -4.8 \cdot 10^{+29}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -3.2 \cdot 10^{-104}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -3.5 \cdot 10^{-178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 1.65 \cdot 10^{-307}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 9.2 \cdot 10^{-219}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-167}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3.2 \cdot 10^{-69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{-33} \lor \neg \left(a \cdot b \leq 1.55 \cdot 10^{+98}\right) \land a \cdot b \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 2
Error	15.8%
Cost	1488

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -5.2 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -4 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 3
Error	48.45%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.25 \cdot 10^{-18}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.05 \cdot 10^{-32} \lor \neg \left(a \cdot b \leq 2.35 \cdot 10^{+98}\right) \land a \cdot b \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 4
Error	25.13%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+102} \lor \neg \left(x \leq 3.8 \cdot 10^{-154}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 5
Error	34.96%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+146}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-96}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6
Error	0.03%
Cost	704

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

Alternative 7
Error	65.42%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023088 
(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)))