Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.2

Time: 6.7s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * 3.0d0)) + (x * (x * (x * (-2.0d0))))
end function

Java

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

↓

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

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

↓

def code(x):
	return (x * (x * 3.0)) + (x * (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(Float64(x * Float64(x * 3.0)) + Float64(x * Float64(x * Float64(x * -2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

↓

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

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

   [Start] 0.2	\[ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   associate-*l* [=>] 0.2	\[ \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{x \cdot \left(3 + x \cdot -2\right)}}} \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	2.1
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	1.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	576

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

Alternative 4
Error	16.8
Cost	320

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

Alternative 5
Error	16.8
Cost	320

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

Alternative 6
Error	61.0
Cost	192

\[x \cdot 4.5 \]

Reproduce

herbie shell --seed 2023039 
(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))))