SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.02% → 0.01%

Time: 5.3s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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);
}

Julia

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

Wolfram

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]

Derivation

1. Initial program 0.02

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

2. Simplified 0.01

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

   [Start] 0.02	\[ x + y \cdot \left(z - x\right) \]
   +-commutative [=>] 0.02	\[ \color{blue}{y \cdot \left(z - x\right) + x} \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(y, z - x, x\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	38.34%
Cost	1180

\[\begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+122}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-60}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-68}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.67%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-68}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	46.37%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-114}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4
Error	1.31%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.145\right):\\ \;\;\;\;y \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5
Error	0.02%
Cost	448

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

Alternative 6
Error	54.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))