Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.29% → 0.27%

Time: 7.7s

Precision: binary64

Cost: 19968

Math

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

↓

\[\frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, {x}^{3} \cdot -4\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (/ x (fma x 2.0 3.0)) (fma x 9.0 (* (pow x 3.0) -4.0))))

C

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

↓

double code(double x) {
	return (x / fma(x, 2.0, 3.0)) * fma(x, 9.0, (pow(x, 3.0) * -4.0));
}

Julia

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

↓

function code(x)
	return Float64(Float64(x / fma(x, 2.0, 3.0)) * fma(x, 9.0, Float64((x ^ 3.0) * -4.0)))
end

Wolfram

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

↓

code[x_] := N[(N[(x / N[(x * 2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(x * 9.0 + N[(N[Power[x, 3.0], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.29%
Target	0.27%
Herbie	0.27%

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

Derivation

1. Initial program 0.29

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

2. Applied egg-rr 6.18

   \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{3 + x \cdot 2}} \]

3. Simplified 0.27

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, {x}^{3} \cdot -4\right)} \]
   Proof

   [Start] 6.18	\[ \frac{\left(x \cdot x\right) \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{3 + x \cdot 2} \]
   associate-/l* [=>] 0.47	\[ \color{blue}{\frac{x \cdot x}{\frac{3 + x \cdot 2}{9 - \left(x \cdot x\right) \cdot 4}}} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{x}{\frac{\frac{3 + x \cdot 2}{9 - \left(x \cdot x\right) \cdot 4}}{x}}} \]
   associate-/r* [<=] 0.36	\[ \frac{x}{\color{blue}{\frac{3 + x \cdot 2}{\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x}}} \]
   associate-/r/ [=>] 0.35	\[ \color{blue}{\frac{x}{3 + x \cdot 2} \cdot \left(\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x\right)} \]
   +-commutative [=>] 0.35	\[ \frac{x}{\color{blue}{x \cdot 2 + 3}} \cdot \left(\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x\right) \]
   fma-def [=>] 0.35	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(x, 2, 3\right)}} \cdot \left(\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x\right) \]
   *-commutative [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \color{blue}{\left(x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)\right)} \]
   sub-neg [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \left(x \cdot \color{blue}{\left(9 + \left(-\left(x \cdot x\right) \cdot 4\right)\right)}\right) \]
   distribute-lft-in [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \color{blue}{\left(x \cdot 9 + x \cdot \left(-\left(x \cdot x\right) \cdot 4\right)\right)} \]
   *-commutative [<=] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \left(x \cdot 9 + \color{blue}{\left(-\left(x \cdot x\right) \cdot 4\right) \cdot x}\right) \]
   fma-def [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \color{blue}{\mathsf{fma}\left(x, 9, \left(-\left(x \cdot x\right) \cdot 4\right) \cdot x\right)} \]
   *-commutative [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, \color{blue}{x \cdot \left(-\left(x \cdot x\right) \cdot 4\right)}\right) \]
   distribute-rgt-neg-in [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(-4\right)\right)}\right) \]
   associate-*r* [=>] 0.35	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(-4\right)}\right) \]
   cube-mult [<=] 0.27	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, \color{blue}{{x}^{3}} \cdot \left(-4\right)\right) \]
   metadata-eval [=>] 0.27	\[ \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, {x}^{3} \cdot \color{blue}{-4}\right) \]

4. Final simplification 0.27

   \[\leadsto \frac{x}{\mathsf{fma}\left(x, 2, 3\right)} \cdot \mathsf{fma}\left(x, 9, {x}^{3} \cdot -4\right) \]

Alternatives

Alternative 1
Error	3.64%
Cost	713

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

Alternative 2
Error	2.87%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\\ \end{array} \]

Alternative 3
Error	2.89%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\frac{-0.5}{x}}\\ \end{array} \]

Alternative 4
Error	0.27%
Cost	704

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

Alternative 5
Error	0.27%
Cost	576

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

Alternative 6
Error	25.31%
Cost	320

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

Alternative 7
Error	25.28%
Cost	320

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

Alternative 8
Error	95.31%
Cost	192

\[x \cdot 4.5 \]

Reproduce

herbie shell --seed 2023090 
(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))))