Main:bigenough2 from A

Average Error: 0.02% → 0.01%

Time: 3.2s

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 0.02

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

2. Simplified 0.01

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + z, x\right)} \]
   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	\[ \color{blue}{\mathsf{fma}\left(y, z + x, x\right)} \]
   +-commutative [=>] 0.01	\[ \mathsf{fma}\left(y, \color{blue}{x + z}, x\right) \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	36.61%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+162}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+54}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+33}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-32}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	19.26%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-28} \lor \neg \left(y \leq 9.5 \cdot 10^{-35}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	1.61%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -34 \lor \neg \left(y \leq 2.5 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 4
Error	37.77%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 54000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	0.02%
Cost	448

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

Alternative 6
Error	54.05%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))