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

Average Error: 0.0 → 0.0

Time: 6.2s

Precision: binary64

Cost: 13248

Math

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

↓

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

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 a b (* t z))))

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(a, b, (t * z)));
}

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(a, b, Float64(t * z)))
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[(a * b + N[(t * z), $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. Taylor expanded in z around 0 0.0

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

4. Simplified 0.0

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

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6976

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

Alternative 2
Error	31.9
Cost	2272

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.22 \cdot 10^{+123}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -580000000000:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-50}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -4.2 \cdot 10^{-134}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;a \cdot b \leq -3.2 \cdot 10^{-219}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 7 \cdot 10^{-111}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;a \cdot b \leq 1.55 \cdot 10^{+57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.65 \cdot 10^{+97}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 3
Error	20.0
Cost	1243

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-14} \lor \neg \left(y \leq 1.5 \cdot 10^{+29}\right) \land \left(y \leq 2.2 \cdot 10^{+34} \lor \neg \left(y \leq 1.4 \cdot 10^{+90}\right) \land \left(y \leq 1.04 \cdot 10^{+105} \lor \neg \left(y \leq 2.3 \cdot 10^{+173}\right)\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot z + a \cdot b\\ \end{array} \]

Alternative 4
Error	16.4
Cost	1240

\[\begin{array}{l} t_1 := t \cdot z + a \cdot b\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	9.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.3 \cdot 10^{+61} \lor \neg \left(a \cdot b \leq 5.3 \cdot 10^{+43}\right):\\ \;\;\;\;t \cdot z + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z + x \cdot y\\ \end{array} \]

Alternative 6
Error	31.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.3 \cdot 10^{+122}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8.5 \cdot 10^{+98}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7
Error	0.0
Cost	704

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

Alternative 8
Error	41.6
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023055 
(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)))