Average Error: 0.0 → 0
Time: 1.3s
Precision: binary64
\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\left(a + b\right) \cdot \left(a + b\right) \]
↓
\[\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(b, 2, a\right)\right) \]
(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))
↓
(FPCore (a b) :precision binary64 (fma b b (* a (fma b 2.0 a))))
double code(double a, double b) {
	return (a + b) * (a + b);
}
↓
double code(double a, double b) {
	return fma(b, b, (a * fma(b, 2.0, a)));
}

function code(a, b)
	return Float64(Float64(a + b) * Float64(a + b))
end
↓
function code(a, b)
	return fma(b, b, Float64(a * fma(b, 2.0, a)))
end

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_] := N[(b * b + N[(a * N[(b * 2.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(b, 2, a\right)\right)

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a \]

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right) \]

  2. Taylor expanded in a around 0 0.0

    \[\leadsto \color{blue}{2 \cdot \left(a \cdot b\right) + \left({a}^{2} + {b}^{2}\right)} \]

  3. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(b, 2, a\right)\right)} \]

  4. Final simplification0

    \[\leadsto \mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(b, 2, a\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))