Expression 2, p15

Average Error: 0.0 → 0

Time: 1.8s

Precision: binary64

Cost: 6592

\[0 \leq x \land x \leq 2\]

Math

\[x + x \cdot x \]

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (fma x x x))

C

double code(double x) {
	return x + (x * x);
}

↓

double code(double x) {
	return fma(x, x, x);
}

Julia

function code(x)
	return Float64(x + Float64(x * x))
end

↓

function code(x)
	return fma(x, x, x)
end

Wolfram

code[x_] := N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * x + x), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0

\[\left(1 + x\right) \cdot x \]

Derivation

1. Initial program 0.0

   \[x + x \cdot x \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x\right)} \]
   Proof

   [Start] 0.0	\[ x + x \cdot x \]
   +-commutative [=>] 0.0	\[ \color{blue}{x \cdot x + x} \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(x, x, x\right)} \]

3. Final simplification 0

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

Alternatives

Alternative 1
Error	0.0
Cost	320

\[x \cdot \left(x + 1\right) \]

Alternative 2
Error	0.0
Cost	320

\[x + x \cdot x \]

Alternative 3
Error	1.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "Expression 2, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :herbie-target
  (* (+ 1.0 x) x)

  (+ x (* x x)))