Average Error: 0.0 → 0.0
Time: 5.8s
Precision: binary64
Cost: 13248
\[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
↓
\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right) \]
(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 (* z t))))
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, (z * t)));
}

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(z * t)))
end

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[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)} \]
    Proof
    (fma.f64 x y (fma.f64 a b (*.f64 z t))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a b) (*.f64 z t)))): 1 points increase in error, 0 points decrease in error
    (fma.f64 x y (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 z t) (*.f64 a b)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 z t) (*.f64 a b)))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost6976
\[\mathsf{fma}\left(z, t, x \cdot y\right) + a \cdot b \]
Alternative 2
Error22.4
Cost1504
\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;b \leq -2.020234640387892 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.450827205969025 \cdot 10^{-268}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 2.7715263207150344 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.020718972946929 \cdot 10^{-258}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 1.2377854063556653 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.0389466000793845 \cdot 10^{-102}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 1.3054603904243763 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.42791705817466535:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error29.5
Cost1492
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.015427151452313654:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.977730114538763 \cdot 10^{-282}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 10^{-322}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.9832583628183144 \cdot 10^{-223}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 3.279320049443658 \cdot 10^{+63}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 4
Error8.7
Cost968
\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -4868435.0340189105:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 4.697526833461144 \cdot 10^{+54}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error8.7
Cost968
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -4868435.0340189105:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 4.697526833461144 \cdot 10^{+54}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]
Alternative 6
Error29.9
Cost712
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.015427151452313654:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.527494712121361 \cdot 10^{+66}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 7
Error0.0
Cost704
\[a \cdot b + \left(z \cdot t + x \cdot y\right) \]
Alternative 8
Error42.1
Cost192
\[a \cdot b \]

Error

Reproduce

herbie shell --seed 2022316 
(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)))