Average Error: 0.0 → 0
Time: 1.6s
Precision: binary64
Cost: 6592
\[x \cdot \left(y + 1\right) \]
↓
\[\mathsf{fma}\left(x, y, x\right) \]
(FPCore (x y) :precision binary64 (* x (+ y 1.0)))
↓
(FPCore (x y) :precision binary64 (fma x y x))
double code(double x, double y) {
	return x * (y + 1.0);
}
↓
double code(double x, double y) {
	return fma(x, y, x);
}

function code(x, y)
	return Float64(x * Float64(y + 1.0))
end
↓
function code(x, y)
	return fma(x, y, x)
end

code[x_, y_] := N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(x * y + x), $MachinePrecision]

x \cdot \left(y + 1\right)
↓
\mathsf{fma}\left(x, y, x\right)

Error

Target

Original0.0
Target0.0
Herbie0
\[x + x \cdot y \]

Derivation

  1. Initial program 0.0

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

  2. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]
    Proof
    (fma.f64 x y x): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (*.f64 x 1))): 3 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y 1))): 6 points increase in error, 2 points decrease in error

  3. Final simplification0

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

Alternatives

Alternative 1
Error1.6
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -64296.49384424692:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.010098926281932824:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[x + x \cdot y \]
Alternative 3
Error27.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, B"
  :precision binary64

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

  (* x (+ y 1.0)))