Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.1

Time: 1.6s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Target

Original	0.2
Target	0.2
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{3 \cdot {x}^{2} - 2 \cdot {x}^{3}} \]

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022150 
(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))))