Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.4% → 100.0%

Time: 3.6s

Alternatives: 3

Speedup: 1.4×

• 27.7% 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: 95.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(x + y\right) \cdot \left(x \cdot 2\right)} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot y \leq 2 \cdot 10^{-321}:\\ \;\;\;\;x \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* x y)) 2e-321) (* x (* y 2.0)) (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * y)) <= 2e-321) {
		tmp = x * (y * 2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * x) + (x * y)) <= 2d-321) then
        tmp = x * (y * 2.0d0)
    else
        tmp = x * (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * y)) <= 2e-321) {
		tmp = x * (y * 2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) + (x * y)) <= 2e-321:
		tmp = x * (y * 2.0)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(x * y)) <= 2e-321)
		tmp = Float64(x * Float64(y * 2.0));
	else
		tmp = Float64(x * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * x) + (x * y)) <= 2e-321)
		tmp = x * (y * 2.0);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], 2e-321], N[(x * N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x x) (*.f64 x y)) < 2.00097e-321

   1. Initial program 100.0%

      \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 2.00097e-321 < (+.f64 (*.f64 x x) (*.f64 x y))

   1. Initial program 91.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 3: 57.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 61.7

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

5. Applied rewrites 61.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 2\right)} \]
6. 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 2024216 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

  (* 2.0 (+ (* x x) (* x y))))