\[x \cdot \left(y + z\right) + z \cdot 5 \]

↓

\[\mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \]

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

↓

(FPCore (x y z) :precision binary64 (fma z 5.0 (* x (+ z y))))

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

↓

double code(double x, double y, double z) {
	return fma(z, 5.0, (x * (z + y)));
}

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

↓

function code(x, y, z)
	return fma(z, 5.0, Float64(x * Float64(z + y)))
end

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(z * 5.0 + N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot \left(y + z\right) + z \cdot 5

↓

\mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right)

Original	0.1
Target	0.1
Herbie	0.0

Derivation

Initial program 0.1
\[x \cdot \left(y + z\right) + z \cdot 5 \]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(x + 5\right)\right)} \]
Proof
(fma.f64 x y (*.f64 z (+.f64 x 5))): 0 points increase in error, 0 points decrease in error
(fma.f64 x y (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 z x) (*.f64 z 5)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 z x) (*.f64 z 5)))): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) (+.f64 (*.f64 z x) (*.f64 z 5))): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 y x) (*.f64 z x)) (*.f64 z 5))): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 x (+.f64 y z))) (*.f64 z 5)): 0 points increase in error, 0 points decrease in error
Taylor expanded in x around 0 0.1
\[\leadsto \color{blue}{\left(y + z\right) \cdot x + 5 \cdot z} \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right)} \]
Proof
(fma.f64 z 5 (*.f64 x (+.f64 z y))): 0 points increase in error, 0 points decrease in error
(fma.f64 z 5 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 y z)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 z 5) (*.f64 x (+.f64 y z)))): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= *-commutative_binary64 (*.f64 5 z)) (*.f64 x (+.f64 y z))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 5 z) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 y z) x))): 0 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (+.f64 y z) x) (*.f64 5 z))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	6848

\[\mathsf{fma}\left(x, y, z \cdot \left(5 + x\right)\right) \]

Alternative 2
Error	13.9
Cost	1115

\[\begin{array}{l} \mathbf{if}\;z \leq -1400 \lor \neg \left(z \leq -2.16 \cdot 10^{-14}\right) \land \left(z \leq -3.6 \cdot 10^{-64} \lor \neg \left(z \leq 2.6 \cdot 10^{-147}\right) \land \left(z \leq 3.6 \cdot 10^{-96} \lor \neg \left(z \leq 10^{+17}\right)\right)\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]

Alternative 3
Error	25.7
Cost	986

\[\begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-14} \lor \neg \left(z \leq -1.6 \cdot 10^{-71}\right) \land \left(z \leq 1.7 \cdot 10^{-121} \lor \neg \left(z \leq 3.7 \cdot 10^{-95}\right) \land z \leq 6.2 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 4
Error	24.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+66}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-66}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 2200000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 5
Error	1.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -124 \lor \neg \left(x \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot 5\\ \end{array} \]

Alternative 6
Error	18.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+155}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Error	0.1
Cost	576

\[x \cdot \left(z + y\right) + z \cdot 5 \]

Alternative 8
Error	35.4
Cost	192

\[z \cdot 5 \]

Error

Target

Derivation

Alternatives

Error

Reproduce