Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.1

Time: 2.2s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (- (* x (* 3.0 x)) (* 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 * (3.0 * x)) - (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 * (3.0d0 * x)) - (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 * (3.0 * x)) - (2.0 * Math.pow(x, 3.0));
}

Python

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

↓

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

Error

Bits error versus x

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

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

3. Taylor expanded in x around 0 0.1

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

4. Applied unpow2_binary64 0.1

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

5. Applied associate-*r*_binary64 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022130 
(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))))