\[\left(1 - x\right) \cdot y + x \cdot z \]

↓

\[\mathsf{fma}\left(x, z - y, y\right) \]

(FPCore (x y z) :precision binary64 (+ (* (- 1.0 x) y) (* x z)))

↓

(FPCore (x y z) :precision binary64 (fma x (- z y) y))

double code(double x, double y, double z) {
	return ((1.0 - x) * y) + (x * z);
}

↓

double code(double x, double y, double z) {
	return fma(x, (z - y), y);
}

function code(x, y, z)
	return Float64(Float64(Float64(1.0 - x) * y) + Float64(x * z))
end

↓

function code(x, y, z)
	return fma(x, Float64(z - y), y)
end

code[x_, y_, z_] := N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x * N[(z - y), $MachinePrecision] + y), $MachinePrecision]

\left(1 - x\right) \cdot y + x \cdot z

↓

\mathsf{fma}\left(x, z - y, y\right)

Original	0.0
Target	0.0
Herbie	0.0

Derivation

Initial program 0.0
\[\left(1 - x\right) \cdot y + x \cdot z \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, z - y, y\right)} \]
Proof
(fma.f64 x (-.f64 z y) y): 0 points increase in error, 0 points decrease in error
(fma.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 z (neg.f64 y))) y): 0 points increase in error, 0 points decrease in error
(fma.f64 x (+.f64 z (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y))) y): 0 points increase in error, 0 points decrease in error
(fma.f64 x (+.f64 z (Rewrite<= *-commutative_binary64 (*.f64 y -1))) y): 0 points increase in error, 0 points decrease in error
(fma.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y -1) z)) y): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 y -1) z)) y)): 1 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x (*.f64 y -1)) (*.f64 x z))) y): 3 points increase in error, 0 points decrease in error
(+.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y -1) x)) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
(+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 -1 x))) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
(+.f64 (+.f64 (*.f64 y (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 y (+.f64 (*.f64 y (neg.f64 x)) (*.f64 x z)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 y (*.f64 y (neg.f64 x))) (*.f64 x z))): 0 points increase in error, 1 points decrease in error
(+.f64 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) (*.f64 y (neg.f64 x))) (*.f64 x z)): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 y (+.f64 1 (neg.f64 x)))) (*.f64 x z)): 3 points increase in error, 2 points decrease in error
(+.f64 (*.f64 y (Rewrite<= sub-neg_binary64 (-.f64 1 x))) (*.f64 x z)): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 x) y)) (*.f64 x z)): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{fma}\left(x, z - y, y\right) \]

Alternatives

Alternative 1
Error	29.9
Cost	1512

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+112}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.3295594800096171 \cdot 10^{+47}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -3.9169128715632526 \cdot 10^{-19}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -2.7202787906946868 \cdot 10^{-185}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -5.717775777224681 \cdot 10^{-251}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.190838602459173 \cdot 10^{-48}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.2467689497507308 \cdot 10^{-16}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+125}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+165}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 2
Error	25.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3801812722991917 \cdot 10^{-121}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.372665985168963 \cdot 10^{-77}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;y \leq 7.72570889896901 \cdot 10^{+55}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	15.9
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(1 - x\right)\\ \mathbf{if}\;y \leq -2.3801812722991917 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.89789429233346 \cdot 10^{-80}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.9
Cost	584

\[\begin{array}{l} t_0 := y + x \cdot z\\ \mathbf{if}\;z \leq -8.769300275088442 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.211919398067523 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	6.9
Cost	584

\[\begin{array}{l} t_0 := y + x \cdot z\\ \mathbf{if}\;z \leq -8.769300275088442 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.211919398067523 \cdot 10^{-77}:\\ \;\;\;\;y - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.1
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -13330597388.302998:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00026611670553742216:\\ \;\;\;\;y + x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.0
Cost	448

\[y + x \cdot \left(z - y\right) \]

Alternative 8
Error	34.9
Cost	64

\[y \]

Error

Target

Derivation

Alternatives

Error

Reproduce