\[\left(x \cdot 3\right) \cdot y - z \]

↓

\[\mathsf{fma}\left(y \cdot x, 3, -z\right) \]

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

↓

(FPCore (x y z) :precision binary64 (fma (* y x) 3.0 (- z)))

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

↓

double code(double x, double y, double z) {
	return fma((y * x), 3.0, -z);
}

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

↓

function code(x, y, z)
	return fma(Float64(y * x), 3.0, Float64(-z))
end

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(y * x), $MachinePrecision] * 3.0 + (-z)), $MachinePrecision]

\left(x \cdot 3\right) \cdot y - z

↓

\mathsf{fma}\left(y \cdot x, 3, -z\right)

Original	0.2
Target	0.1
Herbie	0.1

Derivation

Initial program 0.2
\[\left(x \cdot 3\right) \cdot y - z \]
Simplified0.1
\[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
Proof
(-.f64 (*.f64 x (*.f64 3 y)) z): 0 points increase in error, 0 points decrease in error
(-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 3) y)) z): 18 points increase in error, 10 points decrease in error
Taylor expanded in x around 0 0.1
\[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} - z \]
Applied egg-rr0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, 3, -z\right)} \]
Final simplification0.1
\[\leadsto \mathsf{fma}\left(y \cdot x, 3, -z\right) \]

Alternatives

Alternative 1
Error	23.1
Cost	2140

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;x \cdot 3 \leq -1 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot 3 \leq -2 \cdot 10^{+61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot 3 \leq -5 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot 3 \leq 10^{+69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot 3 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot 3 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot 3 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

\[\left(y \cdot x\right) \cdot 3 - z \]

Alternative 3
Error	0.1
Cost	448

\[x \cdot \left(y \cdot 3\right) - z \]

Alternative 4
Error	26.8
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))

Error

Target

Derivation

Alternatives

Error

Reproduce