Average Error: 0.0 → 0
Time: 1.6s
Precision: binary64
Cost: 6656
\[x \cdot \left(1 - y\right) \]
↓
\[\mathsf{fma}\left(y, -x, x\right) \]
(FPCore (x y) :precision binary64 (* x (- 1.0 y)))
↓
(FPCore (x y) :precision binary64 (fma y (- x) x))
double code(double x, double y) {
	return x * (1.0 - y);
}
↓
double code(double x, double y) {
	return fma(y, -x, x);
}

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

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

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{x - x \cdot y} \]
    Proof
    (-.f64 x (*.f64 x y)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1)) (*.f64 x y)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-lft-out--_binary64 (*.f64 x (-.f64 1 y))): 3 points increase in error, 5 points decrease in error

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{\left(1 - y\right) \cdot x} \]

  4. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, -x, x\right)} \]
    Proof
    (fma.f64 y (neg.f64 x) x): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 x)) x)): 6 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y x))) x): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x))) x): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 -1 y) x)) x): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 -1 y) 1) x)): 3 points increase in error, 5 points decrease in error
    (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1 y))) x): 0 points increase in error, 0 points decrease in error
    (*.f64 (+.f64 1 (Rewrite=> mul-1-neg_binary64 (neg.f64 y))) x): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 y)) x): 0 points increase in error, 0 points decrease in error

  5. Final simplification0

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

Alternatives

Alternative 1
Error1.6
Cost520
\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -64296.49384424692:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.010098926281932824:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[x - y \cdot x \]
Alternative 3
Error0.0
Cost320
\[x \cdot \left(1 - y\right) \]
Alternative 4
Error27.3
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, H"
  :precision binary64
  (* x (- 1.0 y)))