Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Accuracy: 99.7% → 99.8%

Time: 8.3s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return (math.pow(x, 3.0) * -2.0) + (x * (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((x ^ 3.0) * -2.0) + Float64(x * Float64(x * 3.0)))
end

MATLAB

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

↓

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

Target

Original	99.7%
Target	99.8%
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 99.8%

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

   [Start] 99.7	\[ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   associate-*l* [=>] 99.8	\[ \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.8	\[ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]
   sub-neg [=>] 99.8	\[ x \cdot \left(x \cdot \color{blue}{\left(3 + \left(-x \cdot 2\right)\right)}\right) \]
   distribute-lft-in [=>] 99.7	\[ x \cdot \color{blue}{\left(x \cdot 3 + x \cdot \left(-x \cdot 2\right)\right)} \]
   distribute-rgt-neg-in [=>] 99.7	\[ x \cdot \left(x \cdot 3 + x \cdot \color{blue}{\left(x \cdot \left(-2\right)\right)}\right) \]
   metadata-eval [=>] 99.7	\[ x \cdot \left(x \cdot 3 + x \cdot \left(x \cdot \color{blue}{-2}\right)\right) \]

4. Applied egg-rr 99.8%

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

   [Start] 99.7	\[ x \cdot \left(x \cdot 3 + x \cdot \left(x \cdot -2\right)\right) \]
   +-commutative [=>] 99.7	\[ x \cdot \color{blue}{\left(x \cdot \left(x \cdot -2\right) + x \cdot 3\right)} \]
   distribute-lft-in [=>] 99.7	\[ \color{blue}{x \cdot \left(x \cdot \left(x \cdot -2\right)\right) + x \cdot \left(x \cdot 3\right)} \]
   associate-*r* [=>] 99.7	\[ x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot -2\right)} + x \cdot \left(x \cdot 3\right) \]
   associate-*r* [=>] 99.7	\[ \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot -2} + x \cdot \left(x \cdot 3\right) \]
   associate-*l* [<=] 99.7	\[ \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -2 + x \cdot \left(x \cdot 3\right) \]
   pow3 [=>] 99.8	\[ \color{blue}{{x}^{3}} \cdot -2 + x \cdot \left(x \cdot 3\right) \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	96.6%
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
Accuracy	96.5%
Cost	713

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

Alternative 3
Accuracy	99.7%
Cost	704

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

Alternative 4
Accuracy	99.8%
Cost	576

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

Alternative 5
Accuracy	74.2%
Cost	320

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

Alternative 6
Accuracy	74.3%
Cost	320

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

Alternative 7
Accuracy	4.6%
Cost	192

\[x \cdot 4.5 \]

Reproduce

herbie shell --seed 2023147 
(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))))