Average Error: 0.0 → 0.0
Time: 1.7s
Precision: binary64
Cost: 6720
\[0 \leq x \land x \leq 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x \]
↓
\[x \cdot \mathsf{fma}\left(x, x, x\right) \]
(FPCore (x) :precision binary64 (+ (* x (* x x)) (* x x)))
↓
(FPCore (x) :precision binary64 (* x (fma x x x)))
double code(double x) {
	return (x * (x * x)) + (x * x);
}
↓
double code(double x) {
	return x * fma(x, x, x);
}

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

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]

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

Error

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

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x, x\right)} \]
    Proof
    (*.f64 x (fma.f64 x x x)): 0 points increase in error, 0 points decrease in error
    (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) x))): 0 points increase in error, 1 points decrease in error
    (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x (*.f64 x x)) (*.f64 x x))): 4 points increase in error, 1 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost576
\[x \cdot x + x \cdot \left(x \cdot x\right) \]
Alternative 2
Error0.0
Cost448
\[\left(x \cdot x\right) \cdot \left(x + 1\right) \]
Alternative 3
Error1.3
Cost192
\[x \cdot x \]

Error

Reproduce

herbie shell --seed 2022335 
(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)))