Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[0 \leq x \land x \leq 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x \]
↓
\[\mathsf{fma}\left(x, x, {x}^{3}\right) \]
(FPCore (x) :precision binary64 (+ (* x (* x x)) (* x x)))
↓
(FPCore (x) :precision binary64 (fma x x (pow x 3.0)))
double code(double x) {
	return (x * (x * x)) + (x * x);
}
↓
double code(double x) {
	return fma(x, x, pow(x, 3.0));
}

function code(x)
	return Float64(Float64(x * Float64(x * x)) + Float64(x * x))
end
↓
function code(x)
	return fma(x, x, (x ^ 3.0))
end

code[x_] := N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x * x + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(x, x, {x}^{3}\right)

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(1 + x\right) \cdot x\right) \cdot x \]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + {x}^{3}} \]

  4. Applied egg-rr0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, {x}^{3}\right) \]

Reproduce

herbie shell --seed 2022160 
(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)))