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

Average Error: 0.0 → 0.0

Time: 6.5s

Precision: binary64

Cost: 13248

Math

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

↓

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

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 b a (fma x y (* z t))))

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(b, a, fma(x, y, (z * t)));
}

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	30.3
Cost	1752

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -6.289867216669463 \cdot 10^{-27}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq 1.5575458549634238 \cdot 10^{-296}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 7.21851399152233 \cdot 10^{-207}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \cdot a \leq 1.2118992267054724 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 8.502515493648889 \cdot 10^{+19}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \cdot a \leq 7.639544772631747 \cdot 10^{+25}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]

Alternative 2
Error	20.1
Cost	1372

\[\begin{array}{l} t_1 := b \cdot a + z \cdot t\\ \mathbf{if}\;y \leq 3.89552208019858 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.0507984377300053 \cdot 10^{+37}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+82}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+137}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Error	9.4
Cost	968

\[\begin{array}{l} t_1 := b \cdot a + z \cdot t\\ \mathbf{if}\;b \cdot a \leq -6.289867216669463 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 7.639544772631747 \cdot 10^{+25}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	9.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -1.562774054522577 \cdot 10^{-5}:\\ \;\;\;\;b \cdot a + x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 7.639544772631747 \cdot 10^{+25}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;b \cdot a + z \cdot t\\ \end{array} \]

Alternative 5
Error	29.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -1.562774054522577 \cdot 10^{-5}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;b \cdot a \leq 6214156284620.057:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]

Alternative 6
Error	0.0
Cost	704

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

Alternative 7
Error	42.0
Cost	192

\[z \cdot t \]

Reproduce

herbie shell --seed 2022308 
(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)))