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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

Derivation

Initial program 0.0
\[x + \left(y - x\right) \cdot z \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]
Proof
(fma.f64 (-.f64 y x) z x): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) z) x)): 2 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (-.f64 y x) z))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{fma}\left(y - x, z, x\right) \]

Alternatives

Alternative 1
Error	17.5
Cost	848

\[\begin{array}{l} t_0 := x - x \cdot z\\ \mathbf{if}\;x \leq -1.2497137103244571 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3781894118032272 \cdot 10^{-182}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 7.977203018991137 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.640540750021597 \cdot 10^{-44}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.6
Cost	848

\[\begin{array}{l} t_0 := x - x \cdot z\\ t_1 := \left(y - x\right) \cdot z\\ \mathbf{if}\;x \leq -9.065514324971067 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.311728770155945 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.644563165998775 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 342084583155.1468:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	27.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -9.065514324971067 \cdot 10^{+64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.311728770155945 \cdot 10^{-174}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 7.977203018991137 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 342084583155.1468:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	34.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022290 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))

Error

Derivation

Alternatives

Error

Reproduce