Main:bigenough2 from A

Average Error: 0.0 → 0.0

Time: 3.8s

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)): 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

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

Alternatives

Alternative 1
Error	15.7
Cost	1376

\[\begin{array}{l} t_0 := x + y \cdot x\\ t_1 := y \cdot \left(x + z\right)\\ \mathbf{if}\;x \leq -2.703928104080096 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.546836918470571 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.0916380404557543 \cdot 10^{-159}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6198227273489558 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6058251208622737 \cdot 10^{-272}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.0261044064660438 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.826406806563832 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1162337836756516 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 2
Error	17.8
Cost	1112

\[\begin{array}{l} t_0 := x + y \cdot x\\ \mathbf{if}\;x \leq -5.704215073701326 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.546836918470571 \cdot 10^{-134}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq -1.0916380404557543 \cdot 10^{-159}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6198227273489558 \cdot 10^{-295}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 1.6058251208622737 \cdot 10^{-272}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.0261044064660438 \cdot 10^{-146}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.8
Cost	1112

\[\begin{array}{l} \mathbf{if}\;x \leq -5.704215073701326 \cdot 10^{-62}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq -8.546836918470571 \cdot 10^{-134}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq -1.0916380404557543 \cdot 10^{-159}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6198227273489558 \cdot 10^{-295}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 1.6058251208622737 \cdot 10^{-272}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.0261044064660438 \cdot 10^{-146}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4
Error	24.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+101}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1141135824.9214814:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 6.111596886788274 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+44}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	24.5
Cost	456

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

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	34.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))