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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

Derivation

Initial program 0.0
\[x + y \cdot \left(z - x\right) \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(y, z - x, x\right)} \]
Proof
(fma.f64 y (-.f64 z x) x): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (-.f64 z x)) x)): 3 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (-.f64 z x)))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{fma}\left(y, z - x, x\right) \]

Alternatives

Alternative 1
Error	24.0
Cost	1180

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.9 \cdot 10^{+118}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1900000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-55}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 0.0023:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+73}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 7.9 \cdot 10^{+201}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	13.5
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(1 - y\right)\\ t_1 := y \cdot \left(z - x\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.6
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0023:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	24.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 0.0023:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5
Error	0.0
Cost	448

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

Alternative 6
Error	34.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))

Error

Derivation

Alternatives

Error

Reproduce