Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 94.9% → 100.0%

Time: 5.8s

Alternatives: 4

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: 94.9% 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

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

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(Float64(x - y) * x) * 2.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. Final simplification 100.0%

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

Alternative 2: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot -2\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\left(x + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) -2.0)))
   (if (<= y -4.4e+38) t_0 (if (<= y 1.65e-13) (* (+ x x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * x) * -2.0;
	double tmp;
	if (y <= -4.4e+38) {
		tmp = t_0;
	} else if (y <= 1.65e-13) {
		tmp = (x + x) * x;
	} 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 = (y * x) * (-2.0d0)
    if (y <= (-4.4d+38)) then
        tmp = t_0
    else if (y <= 1.65d-13) then
        tmp = (x + x) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * -2.0;
	double tmp;
	if (y <= -4.4e+38) {
		tmp = t_0;
	} else if (y <= 1.65e-13) {
		tmp = (x + x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * -2.0
	tmp = 0
	if y <= -4.4e+38:
		tmp = t_0
	elif y <= 1.65e-13:
		tmp = (x + x) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * -2.0)
	tmp = 0.0
	if (y <= -4.4e+38)
		tmp = t_0;
	elseif (y <= 1.65e-13)
		tmp = Float64(Float64(x + x) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * -2.0;
	tmp = 0.0;
	if (y <= -4.4e+38)
		tmp = t_0;
	elseif (y <= 1.65e-13)
		tmp = (x + x) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[y, -4.4e+38], t$95$0, If[LessEqual[y, 1.65e-13], N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000013e38 or 1.65e-13 < y

   1. Initial program 92.6%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -4.40000000000000013e38 < y < 1.65e-13

   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 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.3%

      \[\leadsto \left(x + x\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+38}:\\ \;\;\;\;\left(y \cdot x\right) \cdot -2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\left(x + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 59.1

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

5. Applied rewrites 59.1%

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

7. Applied rewrites 59.1%

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

9. Applied rewrites 59.1%

   \[\leadsto \left(x + x\right) \cdot x \]
10. Add Preprocessing

Alternative 4: 11.6% 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 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. 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. cancel-sign-sub-inv N/A

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

   10. remove-double-neg N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. pow2 N/A

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

   17. sqrt-pow1 N/A

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

   18. metadata-eval N/A

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

   19. unpow1 N/A

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

   20. distribute-lft-out N/A

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

   21. sqr-neg-rev N/A

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

   22. lift-*.f64 N/A

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

   23. cancel-sub-sign-inv N/A

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

   24. lift-*.f64 N/A

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

4. Applied rewrites 12.3%

   \[\leadsto \color{blue}{0} \]
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 2024298 
(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))))