Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 94.9% → 100.0%

Time: 6.2s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + y\right) \cdot x\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-11}:\\ \;\;\;\;\left(x + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y y) x)))
   (if (<= y -7.6e+45) t_0 (if (<= y 4.3e-11) (* (+ x x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + y) * x;
	double tmp;
	if (y <= -7.6e+45) {
		tmp = t_0;
	} else if (y <= 4.3e-11) {
		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 + y) * x
    if (y <= (-7.6d+45)) then
        tmp = t_0
    else if (y <= 4.3d-11) 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 + y) * x;
	double tmp;
	if (y <= -7.6e+45) {
		tmp = t_0;
	} else if (y <= 4.3e-11) {
		tmp = (x + x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + y) * x
	tmp = 0
	if y <= -7.6e+45:
		tmp = t_0
	elif y <= 4.3e-11:
		tmp = (x + x) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + y) * x)
	tmp = 0.0
	if (y <= -7.6e+45)
		tmp = t_0;
	elseif (y <= 4.3e-11)
		tmp = Float64(Float64(x + x) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + y) * x;
	tmp = 0.0;
	if (y <= -7.6e+45)
		tmp = t_0;
	elseif (y <= 4.3e-11)
		tmp = (x + x) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -7.6e+45], t$95$0, If[LessEqual[y, 4.3e-11], N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.6000000000000004e45 or 4.30000000000000001e-11 < y

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 87.5%

      \[\leadsto \left(y \cdot 2\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 87.5%

       \[\leadsto \left(y + y\right) \cdot x \]

   if -7.6000000000000004e45 < y < 4.30000000000000001e-11

   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

   9. Applied rewrites 89.8%

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

Alternative 3: 56.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y + 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

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(2 \cdot y\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 57.0%

   \[\leadsto \left(y \cdot 2\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 57.0%

    \[\leadsto \left(y + y\right) \cdot x \]
11. 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 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

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. distribute-rgt-out-- N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

   17. sqr-neg-rev N/A

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

   18. pow2 N/A

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

   19. sqrt-pow1 N/A

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

   20. metadata-eval N/A

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

   21. unpow1 N/A

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

   22. distribute-rgt-out-- N/A

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

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, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* x 2) (+ x y)))

  (* 2.0 (+ (* x x) (* x y))))