Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.5% → 100.0%

Time: 3.0s

Alternatives: 3

Speedup: 1.4×

• 28.6% 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: 95.5% 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 95.7%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y + x \cdot x \leq 1.1 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x y) (* x x)) 1.1e-54) (* (* x y) 2.0) (* (* 2.0 x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) + (x * x)) <= 1.1e-54) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = (2.0 * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * y) + (x * x)) <= 1.1d-54) then
        tmp = (x * y) * 2.0d0
    else
        tmp = (2.0d0 * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * y) + (x * x)) <= 1.1e-54) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = (2.0 * x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * y) + (x * x)) <= 1.1e-54:
		tmp = (x * y) * 2.0
	else:
		tmp = (2.0 * x) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) + Float64(x * x)) <= 1.1e-54)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = Float64(Float64(2.0 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * y) + (x * x)) <= 1.1e-54)
		tmp = (x * y) * 2.0;
	else
		tmp = (2.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 1.1e-54], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x x) (*.f64 x y)) < 1.1e-54

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 1.1e-54 < (+.f64 (*.f64 x x) (*.f64 x y))

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   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. lower-*.f64 N/A

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

      4. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + x \cdot x \leq 1.1 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
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. lower-*.f64 N/A

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

   4. lower-*.f64 59.8

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

5. Applied rewrites 59.8%

   \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot x} \]
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 2024244 
(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))))