Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23

Average Error: 0.1% → 0.09%

Time: 9.1s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (* (+ (* x y) z) y) t))

↓

(FPCore (x y z t) :precision binary64 (fma (fma x y z) y t))

C

double code(double x, double y, double z, double t) {
	return (((x * y) + z) * y) + t;
}

↓

double code(double x, double y, double z, double t) {
	return fma(fma(x, y, z), y, t);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * y) + z) * y) + t)
end

↓

function code(x, y, z, t)
	return fma(fma(x, y, z), y, t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(x * y + z), $MachinePrecision] * y + t), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.09

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)} \]
   Proof

   [Start] 0.1	\[ \left(x \cdot y + z\right) \cdot y + t \]
   fma-def [=>] 0.09	\[ \color{blue}{\mathsf{fma}\left(x \cdot y + z, y, t\right)} \]
   fma-def [=>] 0.09	\[ \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, t\right) \]

3. Final simplification 0.09

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

Alternatives

Alternative 1
Error	42.34%
Cost	716

\[\begin{array}{l} t_1 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 3900:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-57} \lor \neg \left(y \leq 12.2\right):\\ \;\;\;\;y \cdot \left(z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot z\\ \end{array} \]

Alternative 3
Error	7.81%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-53} \lor \neg \left(z \leq 95000000000000\right):\\ \;\;\;\;t + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4
Error	21.21%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1700000 \lor \neg \left(y \leq 1.9 \cdot 10^{+153}\right):\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot z\\ \end{array} \]

Alternative 5
Error	0.1%
Cost	576

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

Alternative 6
Error	40.33%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+85}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+141}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 7
Error	46.1%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))