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

Average Error: 0.0 → 0.0

Time: 5.9s

Precision: binary64

Cost: 6976

Math

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

↓

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

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	30.3
Cost	2272

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -4.2 \cdot 10^{-102}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.5 \cdot 10^{-125}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -2.9 \cdot 10^{-160}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.3 \cdot 10^{-281}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 1.18 \cdot 10^{-118}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 4.9 \cdot 10^{-73}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.1 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 2
Error	9.8
Cost	1488

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -2.9 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 5.4 \cdot 10^{-72}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.24 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	23.1
Cost	1243

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-122} \lor \neg \left(y \leq 2.2 \cdot 10^{+32}\right) \land \left(y \leq 1.22 \cdot 10^{+39} \lor \neg \left(y \leq 8 \cdot 10^{+120}\right) \land \left(y \leq 5.6 \cdot 10^{+151} \lor \neg \left(y \leq 2.02 \cdot 10^{+214}\right)\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 4
Error	30.0
Cost	1233

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.3 \cdot 10^{+21}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.18 \cdot 10^{-118} \lor \neg \left(a \cdot b \leq 4 \cdot 10^{-78}\right) \land a \cdot b \leq 3.55 \cdot 10^{-16}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5
Error	13.6
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -9500000000000 \lor \neg \left(x \leq -2.8 \cdot 10^{-17} \lor \neg \left(x \leq -6 \cdot 10^{-48}\right) \land x \leq 2.7 \cdot 10^{-52}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 6
Error	0.0
Cost	704

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

Alternative 7
Error	41.4
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023073 
(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)))