Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Average Error: 0.0 → 0.0

Time: 6.8s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (- (* 2.0 (pow x 2.0)) (* 2.0 (* x y))))

C

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * ((x * x) - (x * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 * (x ** 2.0d0)) - (2.0d0 * (x * y))
end function

Java

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

↓

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

Python

def code(x, y):
	return 2.0 * ((x * x) - (x * y))

↓

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

Julia

function code(x, y)
	return Float64(2.0 * Float64(Float64(x * x) - Float64(x * y)))
end

↓

function code(x, y)
	return Float64(Float64(2.0 * (x ^ 2.0)) - Float64(2.0 * Float64(x * y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (2.0 * (x ^ 2.0)) - (2.0 * (x * y));
end

Wolfram

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

↓

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

Error

Bits error versus x

Bits error versus y

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

4. Applied flip3--_binary64 32.7

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

5. Applied associate-*r/_binary64 32.8

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

6. Applied associate-*r/_binary64 38.7

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

7. Applied add-sqr-sqrt_binary64 38.7

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

8. Applied times-frac_binary64 33.7

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

9. Simplified 33.7

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

10. Simplified 33.7

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

11. Taylor expanded in x around 0 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (* (* x 2.0) (- x y))

  (* 2.0 (- (* x x) (* x y))))