SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.02% → 0.01%

Time: 4.6s

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	31.41%
Cost	1114

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+183}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+27} \lor \neg \left(z \leq -1.02 \cdot 10^{-14}\right) \land \left(z \leq 1.95 \cdot 10^{-14} \lor \neg \left(z \leq 9.7 \cdot 10^{+120}\right) \land z \leq 4.7 \cdot 10^{+160}\right):\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	29.31%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ t_1 := x \cdot \left(1 - y\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.7 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3
Error	47.48%
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+183}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+120}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+160}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4
Error	38.42%
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+72}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5
Error	1.56%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 7.6 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	448

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

Alternative 7
Error	54.32%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))