Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 4.5s

Alternatives: 5

Speedup: 1.4×

• 28.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+140}:\\ \;\;\;\;-2 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-56}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e+140)
   (* -2.0 (* y x))
   (if (<= y 1.4e-56) (* (* x x) 2.0) (* (* -2.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+140) {
		tmp = -2.0 * (y * x);
	} else if (y <= 1.4e-56) {
		tmp = (x * x) * 2.0;
	} else {
		tmp = (-2.0 * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.2d+140)) then
        tmp = (-2.0d0) * (y * x)
    else if (y <= 1.4d-56) then
        tmp = (x * x) * 2.0d0
    else
        tmp = ((-2.0d0) * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+140) {
		tmp = -2.0 * (y * x);
	} else if (y <= 1.4e-56) {
		tmp = (x * x) * 2.0;
	} else {
		tmp = (-2.0 * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e+140:
		tmp = -2.0 * (y * x)
	elif y <= 1.4e-56:
		tmp = (x * x) * 2.0
	else:
		tmp = (-2.0 * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e+140)
		tmp = Float64(-2.0 * Float64(y * x));
	elseif (y <= 1.4e-56)
		tmp = Float64(Float64(x * x) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.2e+140)
		tmp = -2.0 * (y * x);
	elseif (y <= 1.4e-56)
		tmp = (x * x) * 2.0;
	else
		tmp = (-2.0 * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e+140], N[(-2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-56], N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.1999999999999998e140

   1. Initial program 83.3%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if -8.1999999999999998e140 < y < 1.39999999999999997e-56

   1. Initial program 95.5%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.39999999999999997e-56 < y

   1. Initial program 95.6%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 82.8%

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

Alternative 3: 57.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

7. Applied rewrites 56.7%

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

Alternative 4: 57.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

Alternative 5: 11.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 94.1%

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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. lift-*.f64 N/A

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

   7. sqr-neg-rev N/A

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

   8. distribute-lft-out N/A

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

   9. fp-cancel-sub-sign N/A

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

   10. unpow1 N/A

       \[\leadsto \left(x \cdot x - x \cdot y\right) - \color{blue}{{x}^{1}} \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + y\right) \]

   11. metadata-eval N/A

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

   12. sqrt-pow1 N/A

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

   13. pow2 N/A

       \[\leadsto \left(x \cdot x - x \cdot y\right) - \sqrt{\color{blue}{x \cdot x}} \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + y\right) \]

   14. sqr-neg-rev N/A

       \[\leadsto \left(x \cdot x - x \cdot y\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}} \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + y\right) \]

   15. pow2 N/A

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

   16. sqrt-pow1 N/A

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

   17. metadata-eval N/A

       \[\leadsto \left(x \cdot x - x \cdot y\right) - {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}} \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + y\right) \]

   18. unpow1 N/A

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

   19. +-commutative N/A

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

   20. distribute-lft-out N/A

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

   21. sqr-neg-rev N/A

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

   22. lift-*.f64 N/A

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

4. Applied rewrites 9.9%

   \[\leadsto \color{blue}{0} \]
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 2024338 
(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))))