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

Average Error: 0.0 → 0.0

Time: 6.1s

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 0.0

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

2. Simplified 0.0

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

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6976

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

Alternative 2
Error	10.5
Cost	1489

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.9 \cdot 10^{+61}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-86} \lor \neg \left(a \cdot b \leq 3.8 \cdot 10^{-39}\right) \land a \cdot b \leq 1.05 \cdot 10^{+73}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 3
Error	35.9
Cost	1380

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-102}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-159}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-158}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-129}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+47}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+73}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+124}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+136}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Error	31.0
Cost	1233

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.6 \cdot 10^{+27}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-86} \lor \neg \left(a \cdot b \leq 1.8 \cdot 10^{-39}\right) \land a \cdot b \leq 7.5 \cdot 10^{+66}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5
Error	21.2
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-51}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+89} \lor \neg \left(y \leq 2.4 \cdot 10^{+124}\right) \land y \leq 1.25 \cdot 10^{+158}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6
Error	14.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-102} \lor \neg \left(y \leq 2.2 \cdot 10^{+47}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 7
Error	0.0
Cost	704

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

Alternative 8
Error	41.5
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2022356 
(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)))