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

Average Accuracy: 100.0% → 100.0%

Time: 8.0s

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 100.0%

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

2. Simplified 100.0%

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

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

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	43.8%
Cost	1512

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-68}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-206}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-175}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-102}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-58}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+22}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+55}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 10^{+93}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	75.7%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-162} \lor \neg \left(y \leq 8.5 \cdot 10^{-139} \lor \neg \left(y \leq 2 \cdot 10^{+78}\right) \land y \leq 10^{+93}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 4
Accuracy	67.1%
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-32}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+55} \lor \neg \left(y \leq 2.1 \cdot 10^{+78}\right) \land y \leq 1.8 \cdot 10^{+97}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5
Accuracy	84.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{-86} \lor \neg \left(a \cdot b \leq 8.5 \cdot 10^{+30}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]

Alternative 6
Accuracy	52.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.1 \cdot 10^{-89}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7
Accuracy	100.0%
Cost	704

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

Alternative 8
Accuracy	35.0%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023135 
(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)))