Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.1

Time: 16.6s

Precision: binary64

Cost: 13376

Math

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

↓

\[3 \cdot {x}^{2} + -2 \cdot {x}^{3} \]

FPCore

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

↓

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

C

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

↓

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

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 = (3.0d0 * (x ** 2.0d0)) + ((-2.0d0) * (x ** 3.0d0))
end function

Java

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

↓

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

Python

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

↓

def code(x):
	return (3.0 * math.pow(x, 2.0)) + (-2.0 * math.pow(x, 3.0))

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

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

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

   [Start] 0.2	\[ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   rational.json-simplify-2 [=>] 0.2	\[ \color{blue}{\left(3 - x \cdot 2\right) \cdot \left(x \cdot x\right)} \]
   rational.json-simplify-43 [=>] 0.2	\[ \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Taylor expanded in x around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	2.2
Cost	776

\[\begin{array}{l} t_0 := \left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\left(-x\right) \cdot \frac{-3}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.2
Cost	776

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\left(-x\right) \cdot \frac{-3}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-x\right)}{\frac{0.5}{x}}\\ \end{array} \]

Alternative 3
Error	2.2
Cost	712

\[\begin{array}{l} t_0 := \left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(3 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.2
Cost	576

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

Alternative 5
Error	0.2
Cost	576

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

Alternative 6
Error	16.7
Cost	320

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

Alternative 7
Error	16.7
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(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))))