Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Accuracy: 99.7% → 99.8%

Time: 10.0s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return ((x * x) * 3.0) + (-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 = ((x * x) * 3.0d0) + ((-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 ((x * x) * 3.0) + (-2.0 * Math.pow(x, 3.0));
}

Python

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

↓

def code(x):
	return ((x * x) * 3.0) + (-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(Float64(x * x) * 3.0) + 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 = ((x * x) * 3.0) + (-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[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision] + N[(-2.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.7%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Initial program 99.7%

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

2. Simplified 99.7%

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

   [Start] 99.7	\[ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   associate-*l* [=>] 99.7	\[ \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.7	\[ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]
   associate-*r* [=>] 99.7	\[ \color{blue}{\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)} \]
   cancel-sign-sub-inv [=>] 99.7	\[ \left(x \cdot x\right) \cdot \color{blue}{\left(3 + \left(-x\right) \cdot 2\right)} \]
   distribute-lft-in [=>] 99.7	\[ \color{blue}{\left(x \cdot x\right) \cdot 3 + \left(x \cdot x\right) \cdot \left(\left(-x\right) \cdot 2\right)} \]
   distribute-lft-neg-in [<=] 99.7	\[ \left(x \cdot x\right) \cdot 3 + \left(x \cdot x\right) \cdot \color{blue}{\left(-x \cdot 2\right)} \]
   distribute-rgt-neg-in [=>] 99.7	\[ \left(x \cdot x\right) \cdot 3 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(-2\right)\right)} \]
   metadata-eval [=>] 99.7	\[ \left(x \cdot x\right) \cdot 3 + \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{-2}\right) \]

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	96.8%
Cost	713

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

Alternative 2
Accuracy	99.7%
Cost	576

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

Alternative 3
Accuracy	99.7%
Cost	576

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

Alternative 4
Accuracy	74.3%
Cost	320

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

Alternative 5
Accuracy	4.6%
Cost	192

\[x \cdot 4.5 \]

Reproduce

herbie shell --seed 2023135 
(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))))