Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 94.9% → 100.0%

Time: 3.3s

Alternatives: 3

Speedup: 1.4×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \left(x \cdot x - x \cdot y\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(x \cdot x - x \cdot y\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot x\right) \cdot \left(x - y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out-- N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \left(2 \cdot x\right) \cdot \left(x - y\right) \]
6. Add Preprocessing

Alternative 2: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+41}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* -2.0 x) y)))
   (if (<= y -2.9e-19) t_0 (if (<= y 4.5e+41) (* (* x x) 2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (-2.0 * x) * y;
	double tmp;
	if (y <= -2.9e-19) {
		tmp = t_0;
	} else if (y <= 4.5e+41) {
		tmp = (x * x) * 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * x) * y
    if (y <= (-2.9d-19)) then
        tmp = t_0
    else if (y <= 4.5d+41) then
        tmp = (x * x) * 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-2.0 * x) * y;
	double tmp;
	if (y <= -2.9e-19) {
		tmp = t_0;
	} else if (y <= 4.5e+41) {
		tmp = (x * x) * 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-2.0 * x) * y
	tmp = 0
	if y <= -2.9e-19:
		tmp = t_0
	elif y <= 4.5e+41:
		tmp = (x * x) * 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-2.0 * x) * y)
	tmp = 0.0
	if (y <= -2.9e-19)
		tmp = t_0;
	elseif (y <= 4.5e+41)
		tmp = Float64(Float64(x * x) * 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-2.0 * x) * y;
	tmp = 0.0;
	if (y <= -2.9e-19)
		tmp = t_0;
	elseif (y <= 4.5e+41)
		tmp = (x * x) * 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-2.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.9e-19], t$95$0, If[LessEqual[y, 4.5e+41], N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e-19 or 4.5000000000000001e41 < y

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -2.9e-19 < y < 4.5000000000000001e41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-19}:\\ \;\;\;\;\left(-2 \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+41}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2\right) \cdot \left(x - y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* x 2) (- x y)))

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