SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.0 → 0.0

Time: 9.1s

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)): 2 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	23.6
Cost	1180

\[\begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+143}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+51}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -60214154810.830696:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.111596886788274 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	18.3
Cost	1112

\[\begin{array}{l} t_0 := x - y \cdot x\\ \mathbf{if}\;z \leq -8 \cdot 10^{+158}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6.044133337029604 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.503901203901224 \cdot 10^{-49}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.450126118482736 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3
Error	18.3
Cost	1112

\[\begin{array}{l} t_0 := x - y \cdot x\\ \mathbf{if}\;z \leq -8 \cdot 10^{+158}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6.044133337029604 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.503901203901224 \cdot 10^{-49}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.450126118482736 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4
Error	12.0
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -0.0007187408803383929:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.486418110176658 \cdot 10^{-9}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -60214154810.830696:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.560611443828585 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -60214154810.830696:\\ \;\;\;\;y \cdot z - y \cdot x\\ \mathbf{elif}\;y \leq 6.560611443828585 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z - x\right)\\ \end{array} \]

Alternative 7
Error	23.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0007187408803383929:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 6.111596886788274 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 8
Error	0.0
Cost	448

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

Alternative 9
Error	34.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))