SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.02% → 0.01%

Time: 6.2s

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	29.88%
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+181}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{+50}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{elif}\;z \leq 4.15 \cdot 10^{+142}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+189}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	38.43%
Cost	916

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{-36}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-125}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.75%
Cost	849

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-126} \lor \neg \left(y \leq 5.1\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 4
Error	37.92%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-125}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5
Error	1.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -220 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	576

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

Alternative 7
Error	0.02%
Cost	448

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

Alternative 8
Error	54.24%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))