Average Error: 0.0 → 0.0
Time: 5.7s
Precision: binary64
Cost: 13248
\[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
↓
\[\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, z \cdot t\right)\right) \]
(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 a b (fma x y (* 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(a, b, fma(x, y, (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(a, b, fma(x, y, 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[(a * b + N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

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

Alternatives

Alternative 1
Error0.0
Cost6976
\[\mathsf{fma}\left(x, y, z \cdot t\right) + a \cdot b \]
Alternative 2
Error30.5
Cost2272
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+29}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.85 \cdot 10^{-68}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.8 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -7.2 \cdot 10^{-161}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 4.4 \cdot 10^{-306}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 7.1 \cdot 10^{-248}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 3
Error20.8
Cost976
\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+149}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
Alternative 4
Error9.7
Cost968
\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;a \cdot b \leq -13:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 2.7 \cdot 10^{-60}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error30.7
Cost712
\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.45 \cdot 10^{+30}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
Alternative 6
Error14.0
Cost712
\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 21.5:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error0.0
Cost704
\[a \cdot b + \left(z \cdot t + x \cdot y\right) \]
Alternative 8
Error41.6
Cost192
\[a \cdot b \]

Error

Reproduce

herbie shell --seed 2022329 
(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)))