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

Average Error: 0.1 → 0.1

Time: 9.6s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (fma y (fma x y z) 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(y, fma(x, y, z), 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(y, fma(x, y, z), 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[(y * N[(x * y + z), $MachinePrecision] + t), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), t\right)} \]
   Proof
   (fma.f64 y (fma.f64 x y z) t): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) z)) t): 2 points increase in error, 1 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (+.f64 (*.f64 x y) z)) t)): 1 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 x y) z) y)) t): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	13.7
Cost	848

\[\begin{array}{l} t_1 := y \cdot \left(y \cdot x\right)\\ t_2 := t + y \cdot z\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 101385099450620510:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.6
Cost	844

\[\begin{array}{l} t_1 := y \cdot \left(z + y \cdot x\right)\\ \mathbf{if}\;y \leq -1215981697100302000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 151052704120543.6:\\ \;\;\;\;t + y \cdot z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+152}:\\ \;\;\;\;t + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	25.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;t \leq -0.0002179152449677972:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq -1.3413497320467244 \cdot 10^{-79}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;t \leq -4.148290432917078 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.4947624458340494 \cdot 10^{-64}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Error	8.1
Cost	712

\[\begin{array}{l} t_1 := t + y \cdot z\\ \mathbf{if}\;z \leq -9.96863780295364 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.125940662459599 \cdot 10^{+32}:\\ \;\;\;\;t + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.1
Cost	704

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

Alternative 6
Error	0.1
Cost	576

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

Alternative 7
Error	13.6
Cost	320

\[t + y \cdot z \]

Alternative 8
Error	29.9
Cost	64

\[t \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))