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

Average Error: 0.1% → 0.09%

Time: 8.8s

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	43.48%
Cost	1244

\[\begin{array}{l} t_1 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-16}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-43}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-72}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-208}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-242}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.5%
Cost	1241

\[\begin{array}{l} t_1 := t + y \cdot z\\ t_2 := t + x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \left(z + x \cdot y\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-7} \lor \neg \left(t \leq 1.3 \cdot 10^{+89}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	39.64%
Cost	720

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-80}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-165}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-139}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-52}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Error	15.03%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-9} \lor \neg \left(y \leq 6.6 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot \left(z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot z\\ \end{array} \]

Alternative 5
Error	22.72%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+67} \lor \neg \left(y \leq 1.75 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot z\\ \end{array} \]

Alternative 6
Error	0.1%
Cost	576

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

Alternative 7
Error	46.49%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))