Expression 3, p15

Average Error: 0.03% → 0.03%

Time: 2.5s

Precision: binary64

Cost: 6720

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x)
	return Float64(x * fma(x, x, x))
end

Wolfram

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

↓

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

Target

Original	0.03%
Target	0.04%
Herbie	0.03%

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

Derivation

1. Initial program 0.03

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

2. Simplified 0.03

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

   [Start] 0.03	\[ x \cdot \left(x \cdot x\right) + x \cdot x \]
   distribute-lft-out [=>] 0.03	\[ \color{blue}{x \cdot \left(x \cdot x + x\right)} \]
   fma-def [=>] 0.03	\[ x \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)} \]

3. Final simplification 0.03

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

Alternatives

Alternative 1
Error	0.03%
Cost	576

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

Alternative 2
Error	0.04%
Cost	448

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

Alternative 3
Error	2.17%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

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

  (+ (* x (* x x)) (* x x)))