Main:bigenough2 from A

Average Error: 0.02% → 0.01%

Time: 4.1s

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	37.98%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+31}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -13.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+176}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+251}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	19.02%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-42}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.01%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ t_1 := x \cdot \left(y + 1\right)\\ \mathbf{if}\;y \leq -2050000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-44}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 31000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -13.5 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5
Error	36.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	448

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

Alternative 7
Error	54.23%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))