Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.1% → 100.0%

Time: 5.3s

Alternatives: 5

Speedup: 1.6×

• 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: 95.1% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. lift--.f64 N/A

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

   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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-51} \lor \neg \left(y \leq 2250000\right):\\ \;\;\;\;-2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.5e-51) (not (<= y 2250000.0)))
   (* -2.0 (* y x))
   (* (* x 2.0) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e-51) || !(y <= 2250000.0)) {
		tmp = -2.0 * (y * x);
	} else {
		tmp = (x * 2.0) * 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 <= (-4.5d-51)) .or. (.not. (y <= 2250000.0d0))) then
        tmp = (-2.0d0) * (y * x)
    else
        tmp = (x * 2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e-51) || !(y <= 2250000.0)) {
		tmp = -2.0 * (y * x);
	} else {
		tmp = (x * 2.0) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.5e-51) or not (y <= 2250000.0):
		tmp = -2.0 * (y * x)
	else:
		tmp = (x * 2.0) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.5e-51) || !(y <= 2250000.0))
		tmp = Float64(-2.0 * Float64(y * x));
	else
		tmp = Float64(Float64(x * 2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.5e-51) || ~((y <= 2250000.0)))
		tmp = -2.0 * (y * x);
	else
		tmp = (x * 2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.5e-51], N[Not[LessEqual[y, 2250000.0]], $MachinePrecision]], N[(-2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.49999999999999974e-51 or 2.25e6 < y

   1. Initial program 91.4%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

   if -4.49999999999999974e-51 < y < 2.25e6

   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

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   7. Applied rewrites 88.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-51} \lor \neg \left(y \leq 2250000\right):\\ \;\;\;\;-2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% 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 95.3%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 59.4

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

5. Applied rewrites 59.4%

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

7. Applied rewrites 59.4%

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

Alternative 4: 57.6% 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 95.3%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 59.4

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

5. Applied rewrites 59.4%

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

Alternative 5: 12.2% 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 95.3%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. 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 8.7%

   \[\leadsto \color{blue}{0} \]
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 2024320 
(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))))