Average Error: 0.1 → 0.0
Time: 5.4s
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))): 13 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)))): 1 points increase in error, 0 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)): 1 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, 1 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
Error9.6
Cost976
\[\begin{array}{l} t_0 := z + x \cdot 3\\ t_1 := x \cdot 3 + y \cdot 2\\ \mathbf{if}\;x \leq -3.6487534368166766 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.6030247290662018 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.263105406780449 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.514955886308167 \cdot 10^{-8}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error30.7
Cost852
\[\begin{array}{l} \mathbf{if}\;y \leq -2.0845429199297787 \cdot 10^{+142}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -4.7793784402055095 \cdot 10^{+59}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq -268743.9021841809:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -4.939491779292458 \cdot 10^{-40}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 2.206530399144413 \cdot 10^{+84}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 3
Error12.8
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -2.0845429199297787 \cdot 10^{+142}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 8.16107697805488 \cdot 10^{+143}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 4
Error9.7
Cost584
\[\begin{array}{l} t_0 := z + x \cdot 3\\ \mathbf{if}\;x \leq -5.263105406780449 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6463058587838982 \cdot 10^{-6}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error0.1
Cost576
\[x + \left(z + 2 \cdot \left(x + y\right)\right) \]
Alternative 6
Error30.8
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -5.263105406780449 \cdot 10^{+98}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 1.514955886308167 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 7
Error41.9
Cost64
\[z \]

Error

Reproduce

herbie shell --seed 2022291 
(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))