Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 94.7% → 100.0%

Time: 2.9s

Alternatives: 3

Speedup: 1.3×

• 28.6% 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: 94.7% 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 95.3%

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

   1. *-commutative 95.3%

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

   2. distribute-lft-out-- 100.0%

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

   3. associate-*l* 100.0%

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

   4. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+49} \lor \neg \left(y \leq -7.6 \cdot 10^{+23}\right) \land \left(y \leq -1.96 \cdot 10^{-11} \lor \neg \left(y \leq 5.2 \cdot 10^{-53}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.3e+49)
         (and (not (<= y -7.6e+23))
              (or (<= y -1.96e-11) (not (<= y 5.2e-53)))))
   (* y (* x -2.0))
   (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.3e+49) || (!(y <= -7.6e+23) && ((y <= -1.96e-11) || !(y <= 5.2e-53)))) {
		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 <= (-2.3d+49)) .or. (.not. (y <= (-7.6d+23))) .and. (y <= (-1.96d-11)) .or. (.not. (y <= 5.2d-53))) 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 <= -2.3e+49) || (!(y <= -7.6e+23) && ((y <= -1.96e-11) || !(y <= 5.2e-53)))) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.3e+49) or (not (y <= -7.6e+23) and ((y <= -1.96e-11) or not (y <= 5.2e-53))):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.3e+49) || (!(y <= -7.6e+23) && ((y <= -1.96e-11) || !(y <= 5.2e-53))))
		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 <= -2.3e+49) || (~((y <= -7.6e+23)) && ((y <= -1.96e-11) || ~((y <= 5.2e-53)))))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.3e+49], And[N[Not[LessEqual[y, -7.6e+23]], $MachinePrecision], Or[LessEqual[y, -1.96e-11], N[Not[LessEqual[y, 5.2e-53]], $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 < -2.30000000000000002e49 or -7.5999999999999995e23 < y < -1.95999999999999989e-11 or 5.19999999999999993e-53 < y

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 86.8%

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

      1. associate-*r* 86.8%

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

   4. Simplified 86.8%

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

   if -2.30000000000000002e49 < y < -7.5999999999999995e23 or -1.95999999999999989e-11 < y < 5.19999999999999993e-53

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

      2. unpow2 90.4%

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

      3. associate-*l* 90.4%

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

   4. Simplified 90.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+49} \lor \neg \left(y \leq -7.6 \cdot 10^{+23}\right) \land \left(y \leq -1.96 \cdot 10^{-11} \lor \neg \left(y \leq 5.2 \cdot 10^{-53}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \]

Alternative 3: 58.6% 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 95.3%

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

2. Taylor expanded in x around inf 57.2%

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

   1. *-commutative 57.2%

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

   2. unpow2 57.2%

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

   3. associate-*l* 57.2%

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

4. Simplified 57.2%

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

5. Final simplification 57.2%

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

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 2023297 
(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))))