Average Error: 0.1 → 0.0
Time: 9.7s
Precision: binary64
Cost: 13120
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
↓
\[\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right) \]
(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))
↓
(FPCore (x y z) :precision binary64 (fma x 3.0 (fma y 2.0 z)))
double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}
↓
double code(double x, double y, double z) {
	return fma(x, 3.0, fma(y, 2.0, z));
}

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end
↓
function code(x, y, z)
	return fma(x, 3.0, fma(y, 2.0, z))
end

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]
↓
code[x_, y_, z_] := N[(x * 3.0 + N[(y * 2.0 + z), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)

Error

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]
    Proof
    (fma.f64 x 3 (fma.f64 y 2 z)): 0 points increase in error, 0 points decrease in error
    (fma.f64 x (Rewrite<= metadata-eval (+.f64 2 1)) (fma.f64 y 2 z)): 0 points increase in error, 0 points decrease in error
    (fma.f64 x (+.f64 2 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y 2) z))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x (+.f64 2 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 y)) z)): 0 points increase in error, 0 points decrease in error
    (fma.f64 x (+.f64 2 1) (+.f64 (Rewrite<= count-2_binary64 (+.f64 y y)) z)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 2 1)) (+.f64 (+.f64 y y) z))): 16 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 2 1) x)) (+.f64 (+.f64 y y) z)): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 2 x))) (+.f64 (+.f64 y y) z)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 (*.f64 2 x) (+.f64 (+.f64 y y) z)))): 2 points increase in error, 3 points decrease in error
    (+.f64 x (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 x) (+.f64 y y)) z))): 0 points increase in error, 0 points decrease in error
    (+.f64 x (+.f64 (+.f64 (Rewrite<= count-2_binary64 (+.f64 x x)) (+.f64 y y)) z)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 x (+.f64 y y)))) z)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (+.f64 (+.f64 x (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x y) y))) z)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 x y) y) x)) z)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 x y) y) x) z) x)): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right) \]

Alternatives

Alternative 1
Error30.4
Cost1116
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1223570910276925 \cdot 10^{+77}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -3.530529520878461 \cdot 10^{-52}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -3.968969318656078 \cdot 10^{-204}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.341687256356038 \cdot 10^{-283}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.0365057400262258 \cdot 10^{-248}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.976540680210769 \cdot 10^{-186}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.744926158905402 \cdot 10^{+90}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 2
Error9.0
Cost712
\[\begin{array}{l} t_0 := x + 2 \cdot \left(x + y\right)\\ \mathbf{if}\;y \leq -1.022256213092302 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.923805451879806 \cdot 10^{+28}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error9.0
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq -1.022256213092302 \cdot 10^{+64}:\\ \;\;\;\;y \cdot 2 + x \cdot 3\\ \mathbf{elif}\;y \leq 1.923805451879806 \cdot 10^{+28}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]
Alternative 4
Error9.0
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq -1.022256213092302 \cdot 10^{+64}:\\ \;\;\;\;y \cdot 2 + x \cdot 3\\ \mathbf{elif}\;y \leq 1.923805451879806 \cdot 10^{+28}:\\ \;\;\;\;x + \left(z + \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]
Alternative 5
Error12.6
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1452077813156415 \cdot 10^{+105}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 8.680625368788571 \cdot 10^{+70}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 6
Error9.6
Cost584
\[\begin{array}{l} t_0 := z + x \cdot 3\\ \mathbf{if}\;x \leq -119233.22702540075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.744926158905402 \cdot 10^{+90}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error0.1
Cost576
\[x + \left(z + 2 \cdot \left(x + y\right)\right) \]
Alternative 8
Error30.7
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1223570910276925 \cdot 10^{+77}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 2.4897210683865577 \cdot 10^{-79}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 9
Error41.6
Cost64
\[z \]

Error

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))