Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.4% → 100.0%

Time: 2.6s

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: 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

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

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 93.3%

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

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot 2\\ \mathbf{if}\;y \leq -3 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x y) 2.0)))
   (if (<= y -3e-18) t_0 (if (<= y 4.5e+72) (* (* x x) 2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * y) * 2.0;
	double tmp;
	if (y <= -3e-18) {
		tmp = t_0;
	} else if (y <= 4.5e+72) {
		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 = (x * y) * 2.0d0
    if (y <= (-3d-18)) then
        tmp = t_0
    else if (y <= 4.5d+72) 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 = (x * y) * 2.0;
	double tmp;
	if (y <= -3e-18) {
		tmp = t_0;
	} else if (y <= 4.5e+72) {
		tmp = (x * x) * 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) * 2.0
	tmp = 0
	if y <= -3e-18:
		tmp = t_0
	elif y <= 4.5e+72:
		tmp = (x * x) * 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * 2.0)
	tmp = 0.0
	if (y <= -3e-18)
		tmp = t_0;
	elseif (y <= 4.5e+72)
		tmp = Float64(Float64(x * x) * 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) * 2.0;
	tmp = 0.0;
	if (y <= -3e-18)
		tmp = t_0;
	elseif (y <= 4.5e+72)
		tmp = (x * x) * 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[y, -3e-18], t$95$0, If[LessEqual[y, 4.5e+72], N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.99999999999999983e-18 or 4.4999999999999998e72 < y

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -2.99999999999999983e-18 < y < 4.4999999999999998e72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 1.7× speedup

Math

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

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(Float64(x * x) * 2.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 57.5

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

5. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 2} \]
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 2024295 
(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))))