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

Average Error: 0.03% → 0.02%

Time: 5.3s

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 0.02

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, a \cdot b\right)\right)} \]
   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 [=>] 0.02	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

3. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.03%
Cost	6976

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

Alternative 2
Error	48.63%
Cost	2012

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.05 \cdot 10^{+66}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -4.3 \cdot 10^{-141}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -7.2 \cdot 10^{-189}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.2 \cdot 10^{-272}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-317}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.08 \cdot 10^{-100}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 0.235:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 3
Error	16.05%
Cost	1228

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -1.25 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 1.7 \cdot 10^{-73}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.5 \cdot 10^{+152}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.88%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+140} \lor \neg \left(x \leq -8.6 \cdot 10^{+81} \lor \neg \left(x \leq -1.55 \cdot 10^{-36}\right) \land x \leq 3.7 \cdot 10^{-93}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 5
Error	48.47%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 0.125:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 6
Error	30.06%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+141}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+35}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Error	0.03%
Cost	704

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

Alternative 8
Error	66.02%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023102 
(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)))