Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 4.3s

Precision: binary64

Cost: 6976

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]

↓

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, 3, x \cdot \left(x \cdot -2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

↓

(FPCore (x) :precision binary64 (* x (fma x 3.0 (* x (* x -2.0)))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

↓

double code(double x) {
	return x * fma(x, 3.0, (x * (x * -2.0)));
}

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

↓

function code(x)
	return Float64(x * fma(x, 3.0, Float64(x * Float64(x * -2.0))))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * 3.0 + N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.8%
Target	99.8%
Herbie	99.8%

\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]

2. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   Step-by-step derivation

   [Start] 99.8%	\[ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   associate-*l* [=>] 99.9%	\[ \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Applied egg-rr 99.9%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 3, x \cdot \left(x \cdot -2\right)\right)} \]
   Step-by-step derivation

   [Start] 99.9%	\[ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]
   sub-neg [=>] 99.9%	\[ x \cdot \left(x \cdot \color{blue}{\left(3 + \left(-x \cdot 2\right)\right)}\right) \]
   distribute-lft-in [=>] 99.8%	\[ x \cdot \color{blue}{\left(x \cdot 3 + x \cdot \left(-x \cdot 2\right)\right)} \]
   fma-def [=>] 99.9%	\[ x \cdot \color{blue}{\mathsf{fma}\left(x, 3, x \cdot \left(-x \cdot 2\right)\right)} \]
   distribute-rgt-neg-in [=>] 99.9%	\[ x \cdot \mathsf{fma}\left(x, 3, x \cdot \color{blue}{\left(x \cdot \left(-2\right)\right)}\right) \]
   metadata-eval [=>] 99.9%	\[ x \cdot \mathsf{fma}\left(x, 3, x \cdot \left(x \cdot \color{blue}{-2}\right)\right) \]

4. Final simplification 99.9%

   \[\leadsto x \cdot \mathsf{fma}\left(x, 3, x \cdot \left(x \cdot -2\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6976

\[x \cdot \mathsf{fma}\left(x, 3, x \cdot \left(x \cdot -2\right)\right) \]

Alternative 2
Accuracy	98.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	576

\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Alternative 4
Accuracy	61.4%
Cost	320

\[3 \cdot \left(x \cdot x\right) \]

Alternative 5
Accuracy	3.1%
Cost	192

\[x \cdot 4.5 \]

Alternative 6
Accuracy	2.5%
Cost	64

\[-6.75 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

  (* (* x x) (- 3.0 (* x 2.0))))