Main:bigenough2 from A

Average Error: 0.0 → 0.0

Time: 3.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma y (+ x z) 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, (x + z), 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(x + z), x)
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(y * N[(x + z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + z, x\right)} \]
   Proof
   (fma.f64 y (+.f64 x z) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (Rewrite<= +-commutative_binary64 (+.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)): 5 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

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

Alternatives

Alternative 1
Error	24.0
Cost	1116

\[\begin{array}{l} \mathbf{if}\;y \leq -2.655558862252027 \cdot 10^{+21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.37150999400213 \cdot 10^{-27}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.826276484937989 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.8616239359329024 \cdot 10^{-129}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 4.714813121874414 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	11.9
Cost	716

\[\begin{array}{l} t_0 := x + y \cdot z\\ \mathbf{if}\;y \leq -2.655558862252027 \cdot 10^{+21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+133}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.1
Cost	584

\[\begin{array}{l} t_0 := x + y \cdot z\\ \mathbf{if}\;z \leq -1.1912780477687877 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.050232470255792 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.8
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ \mathbf{if}\;y \leq -7.095618448588677:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.05060558891981524:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -7.095618448588677:\\ \;\;\;\;y \cdot z + y \cdot x\\ \mathbf{elif}\;y \leq 0.05060558891981524:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]

Alternative 6
Error	0.0
Cost	576

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

Alternative 7
Error	24.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0283923311559037:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.05060558891981524:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	0.0
Cost	448

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

Alternative 9
Error	35.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))