Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.2

Time: 7.6s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x)
	return Float64(x * Float64(x * fma(x, -2.0, 3.0)))
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 * N[(x * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \]
   Proof
   (*.f64 x (*.f64 x (fma.f64 x -2 3))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (fma.f64 x (Rewrite<= metadata-eval (neg.f64 2)) 3))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (neg.f64 2)) 3)))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 2))) 3))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x) 2)) 3))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 3 (*.f64 (neg.f64 x) 2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 x (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 3 (*.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) (-.f64 3 (*.f64 x 2)))): 27 points increase in error, 16 points decrease in error

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	2.4
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{if}\;x \leq -2543981135337.477:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.370840529417722:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

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

Alternative 3
Error	0.2
Cost	576

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

Alternative 4
Error	16.1
Cost	320

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

Alternative 5
Error	61.0
Cost	192

\[x \cdot 4.5 \]

Alternative 6
Error	61.4
Cost	64

\[6.75 \]

Reproduce

herbie shell --seed 2022297 
(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))))