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

Average Error: 0.1 → 0.1

Time: 5.2s

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 0.1

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.9
Cost	976

\[\begin{array}{l} t_1 := t + y \cdot z\\ t_2 := y \cdot \left(z + x \cdot y\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.021:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 2
Error	24.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-90}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-70}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 0.021:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 3
Error	9.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00053 \lor \neg \left(y \leq 9.2 \cdot 10^{-18}\right):\\ \;\;\;\;y \cdot \left(z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot z\\ \end{array} \]

Alternative 4
Error	4.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-5} \lor \neg \left(z \leq 5.2 \cdot 10^{-37}\right):\\ \;\;\;\;t + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 5
Error	14.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+15} \lor \neg \left(y \leq 2.9 \cdot 10^{+110}\right):\\ \;\;\;\;t + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 6
Error	0.1
Cost	576

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

Alternative 7
Error	24.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-86}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-70}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 8
Error	30.0
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))