SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.0 → 0.0

Time: 8.0s

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 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, z - x, x\right)} \]
   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)): 1 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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	25.1
Cost	1576

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+58}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -241906161300961.78:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.932988695682121 \cdot 10^{-74}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.3772021743536337 \cdot 10^{-220}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.894651963428366 \cdot 10^{-210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.173746397033535 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.355473785437023 \cdot 10^{-57}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 7.702306877238973 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 511715009402941:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	16.8
Cost	1112

\[\begin{array}{l} t_0 := x - y \cdot x\\ \mathbf{if}\;x \leq -1.21152117622049 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.368045642533995 \cdot 10^{-185}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq -2.851695269204857 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.131852731387771 \cdot 10^{-68}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 191298760491899500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.899743910600573 \cdot 10^{+36}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.8
Cost	1112

\[\begin{array}{l} t_0 := x - y \cdot x\\ \mathbf{if}\;x \leq -1.21152117622049 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.368045642533995 \cdot 10^{-185}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq -2.851695269204857 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.131852731387771 \cdot 10^{-68}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 191298760491899500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.899743910600573 \cdot 10^{+36}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 4
Error	13.5
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -2.932988695682121 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3772021743536337 \cdot 10^{-220}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.894651963428366 \cdot 10^{-210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.173746397033535 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.355473785437023 \cdot 10^{-57}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 7.702306877238973 \cdot 10^{-13}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	24.9
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -2.932988695682121 \cdot 10^{-74}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.3772021743536337 \cdot 10^{-220}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.894651963428366 \cdot 10^{-210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.173746397033535 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.355473785437023 \cdot 10^{-57}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 7.702306877238973 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 6
Error	1.3
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -241906161300961.78:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.702306877238973 \cdot 10^{-13}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.0
Cost	448

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

Alternative 8
Error	35.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))