Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.3% → 100.0%

Time: 4.9s

Alternatives: 5

Speedup: 1.3×

• 28.0% 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: 95.3% 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, 1.3× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot -1\right)} \cdot y + 2 \cdot x\right) \]

   2. associate-*r* 100.0%

      \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(-1 \cdot y\right)} + 2 \cdot x\right) \]

   3. neg-mul-1 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. +-commutative 100.0%

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

   8. sub-neg 100.0%

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

   9. associate-*r* 100.0%

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

   10. *-commutative 100.0%

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

7. Simplified 100.0%

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

Alternative 2: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-21} \lor \neg \left(y \leq 0.00145\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e-21) (not (<= y 0.00145)))
   (* y (* x -2.0))
   (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e-21) || !(y <= 0.00145)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.2d-21)) .or. (.not. (y <= 0.00145d0))) then
        tmp = y * (x * (-2.0d0))
    else
        tmp = x * (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e-21) || !(y <= 0.00145)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2e-21) or not (y <= 0.00145):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e-21) || !(y <= 0.00145))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(x * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.2e-21) || ~((y <= 0.00145)))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e-21], N[Not[LessEqual[y, 0.00145]], $MachinePrecision]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e-21 or 0.00145 < y

   1. Initial program 90.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. associate-*r* 82.8%

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

   7. Simplified 82.8%

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

   if -1.2e-21 < y < 0.00145

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot -1\right)} \cdot y + 2 \cdot x\right) \]

      2. associate-*r* 100.0%

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(-1 \cdot y\right)} + 2 \cdot x\right) \]

      3. neg-mul-1 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 87.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-21} \lor \neg \left(y \leq 0.00145\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

Alternative 4: 58.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot -1\right)} \cdot y + 2 \cdot x\right) \]

   2. associate-*r* 100.0%

      \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(-1 \cdot y\right)} + 2 \cdot x\right) \]

   3. neg-mul-1 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. +-commutative 100.0%

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

   8. sub-neg 100.0%

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

   9. associate-*r* 100.0%

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

   10. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 56.1%

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

Alternative 5: 58.0% accurate, 1.8× speedup

Math

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

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(2.0 * Float64(x * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 56.1%

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

Developer Target 1: 100.0% accurate, 1.3× 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 2024121 
(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))))