Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.6% → 100.0%

Time: 2.5s

Alternatives: 4

Speedup: 1.3×

• 28.7% 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: 95.6% 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

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. Taylor expanded in x around 0 96.1%

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

   1. metadata-eval 96.1%

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

   2. associate-*r* 96.1%

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

   3. mul-1-neg 96.1%

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

   4. distribute-rgt-neg-out 96.1%

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

   5. unpow2 96.1%

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

   6. distribute-lft-in 96.1%

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

   7. +-commutative 96.1%

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

   8. distribute-lft-out 100.0%

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

   9. sub-neg 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*r* 100.0%

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

   12. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+56} \lor \neg \left(y \leq 1.3 \cdot 10^{-24}\right) \land \left(y \leq 4100000000 \lor \neg \left(y \leq 4.5 \cdot 10^{+30}\right)\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6e+56)
         (and (not (<= y 1.3e-24))
              (or (<= y 4100000000.0) (not (<= y 4.5e+30)))))
   (* -2.0 (* x y))
   (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6e+56) || (!(y <= 1.3e-24) && ((y <= 4100000000.0) || !(y <= 4.5e+30)))) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = x * (x + 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 <= (-6d+56)) .or. (.not. (y <= 1.3d-24)) .and. (y <= 4100000000.0d0) .or. (.not. (y <= 4.5d+30))) then
        tmp = (-2.0d0) * (x * y)
    else
        tmp = x * (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6e+56) || (!(y <= 1.3e-24) && ((y <= 4100000000.0) || !(y <= 4.5e+30)))) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6e+56) or (not (y <= 1.3e-24) and ((y <= 4100000000.0) or not (y <= 4.5e+30))):
		tmp = -2.0 * (x * y)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6e+56) || (!(y <= 1.3e-24) && ((y <= 4100000000.0) || !(y <= 4.5e+30))))
		tmp = Float64(-2.0 * Float64(x * y));
	else
		tmp = Float64(x * Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6e+56) || (~((y <= 1.3e-24)) && ((y <= 4100000000.0) || ~((y <= 4.5e+30)))))
		tmp = -2.0 * (x * y);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6e+56], And[N[Not[LessEqual[y, 1.3e-24]], $MachinePrecision], Or[LessEqual[y, 4100000000.0], N[Not[LessEqual[y, 4.5e+30]], $MachinePrecision]]]], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000012e56 or 1.3e-24 < y < 4.1e9 or 4.49999999999999995e30 < y

   1. Initial program 92.8%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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. Taylor expanded in x around 0 90.9%

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

   if -6.00000000000000012e56 < y < 1.3e-24 or 4.1e9 < y < 4.49999999999999995e30

   1. Initial program 99.2%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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. Taylor expanded in x around inf 90.9%

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 90.9%

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

      2. associate-*r* 90.9%

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

      3. *-commutative 90.9%

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

      4. count-2 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+56} \lor \neg \left(y \leq 1.3 \cdot 10^{-24}\right) \land \left(y \leq 4100000000 \lor \neg \left(y \leq 4.5 \cdot 10^{+30}\right)\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

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 96.1%

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

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

Alternative 4: 56.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. Taylor expanded in x around 0 59.9%

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

5. Final simplification 59.9%

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

Developer target: 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 2023293 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (* (* x 2.0) (- x y))

  (* 2.0 (- (* x x) (* x y))))