Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.2

Time: 6.6s

Precision: binary64

Cost: 7680

Math

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

↓

\[\frac{27 - {x}^{3} \cdot 8}{9 + x \cdot \left(x \cdot 4 + 6\right)} \cdot \left(x \cdot x\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (/ (- 27.0 (* (pow x 3.0) 8.0)) (+ 9.0 (* x (+ (* x 4.0) 6.0)))) (* x x)))

C

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

↓

double code(double x) {
	return ((27.0 - (pow(x, 3.0) * 8.0)) / (9.0 + (x * ((x * 4.0) + 6.0)))) * (x * x);
}

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 = ((27.0d0 - ((x ** 3.0d0) * 8.0d0)) / (9.0d0 + (x * ((x * 4.0d0) + 6.0d0)))) * (x * x)
end function

Java

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

↓

public static double code(double x) {
	return ((27.0 - (Math.pow(x, 3.0) * 8.0)) / (9.0 + (x * ((x * 4.0) + 6.0)))) * (x * x);
}

Python

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

↓

def code(x):
	return ((27.0 - (math.pow(x, 3.0) * 8.0)) / (9.0 + (x * ((x * 4.0) + 6.0)))) * (x * x)

Julia

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

↓

function code(x)
	return Float64(Float64(Float64(27.0 - Float64((x ^ 3.0) * 8.0)) / Float64(9.0 + Float64(x * Float64(Float64(x * 4.0) + 6.0)))) * Float64(x * x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((27.0 - ((x ^ 3.0) * 8.0)) / (9.0 + (x * ((x * 4.0) + 6.0)))) * (x * x);
end

Wolfram

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

↓

code[x_] := N[(N[(N[(27.0 - N[(N[Power[x, 3.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] / N[(9.0 + N[(x * N[(N[(x * 4.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $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. Applied egg-rr 6.3

   \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(27 - {\left(x \cdot 2\right)}^{3}\right)}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{\frac{27 - {x}^{3} \cdot 8}{9 + x \cdot \left(x \cdot 4 + 6\right)} \cdot \left(x \cdot x\right)} \]
   Proof

   [Start] 6.3	\[ \frac{\left(x \cdot x\right) \cdot \left(27 - {\left(x \cdot 2\right)}^{3}\right)}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)} \]
   *-commutative [<=] 6.3	\[ \frac{\color{blue}{\left(27 - {\left(x \cdot 2\right)}^{3}\right) \cdot \left(x \cdot x\right)}}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)} \]
   associate-*l/ [<=] 0.2	\[ \color{blue}{\frac{27 - {\left(x \cdot 2\right)}^{3}}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)} \cdot \left(x \cdot x\right)} \]
   cube-prod [=>] 0.2	\[ \frac{27 - \color{blue}{{x}^{3} \cdot {2}^{3}}}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)} \cdot \left(x \cdot x\right) \]
   metadata-eval [=>] 0.2	\[ \frac{27 - {x}^{3} \cdot \color{blue}{8}}{9 + \left(\left(x \cdot x\right) \cdot 4 + 6 \cdot x\right)} \cdot \left(x \cdot x\right) \]
   associate-*l* [=>] 0.2	\[ \frac{27 - {x}^{3} \cdot 8}{9 + \left(\color{blue}{x \cdot \left(x \cdot 4\right)} + 6 \cdot x\right)} \cdot \left(x \cdot x\right) \]
   *-commutative [=>] 0.2	\[ \frac{27 - {x}^{3} \cdot 8}{9 + \left(x \cdot \left(x \cdot 4\right) + \color{blue}{x \cdot 6}\right)} \cdot \left(x \cdot x\right) \]
   distribute-lft-out [=>] 0.2	\[ \frac{27 - {x}^{3} \cdot 8}{9 + \color{blue}{x \cdot \left(x \cdot 4 + 6\right)}} \cdot \left(x \cdot x\right) \]

4. Final simplification 0.2

   \[\leadsto \frac{27 - {x}^{3} \cdot 8}{9 + x \cdot \left(x \cdot 4 + 6\right)} \cdot \left(x \cdot x\right) \]

Alternatives

Alternative 1
Error	2.2
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	0.2
Cost	576

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

Alternative 3
Error	16.1
Cost	320

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

Alternative 4
Error	16.1
Cost	320

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

Alternative 5
Error	61.0
Cost	192

\[x \cdot 4.5 \]

Reproduce

herbie shell --seed 2023054 
(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))))