Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 1.9s

Alternatives: 3

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: 95.4% 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 95.7%

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

   1. distribute-lft-out-- 100.0%

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

   2. associate-*r* 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-34} \lor \neg \left(y \leq 4.5 \cdot 10^{-11}\right) \land \left(y \leq 4.4 \cdot 10^{+100} \lor \neg \left(y \leq 1.3 \cdot 10^{+126}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.5e-34)
         (and (not (<= y 4.5e-11)) (or (<= y 4.4e+100) (not (<= y 1.3e+126)))))
   (* x (* y -2.0))
   (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.5e-34) || (!(y <= 4.5e-11) && ((y <= 4.4e+100) || !(y <= 1.3e+126)))) {
		tmp = x * (y * -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 <= (-9.5d-34)) .or. (.not. (y <= 4.5d-11)) .and. (y <= 4.4d+100) .or. (.not. (y <= 1.3d+126))) then
        tmp = x * (y * (-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 <= -9.5e-34) || (!(y <= 4.5e-11) && ((y <= 4.4e+100) || !(y <= 1.3e+126)))) {
		tmp = x * (y * -2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.5e-34) or (not (y <= 4.5e-11) and ((y <= 4.4e+100) or not (y <= 1.3e+126))):
		tmp = x * (y * -2.0)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.5e-34) || (!(y <= 4.5e-11) && ((y <= 4.4e+100) || !(y <= 1.3e+126))))
		tmp = Float64(x * Float64(y * -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 <= -9.5e-34) || (~((y <= 4.5e-11)) && ((y <= 4.4e+100) || ~((y <= 1.3e+126)))))
		tmp = x * (y * -2.0);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.5e-34], And[N[Not[LessEqual[y, 4.5e-11]], $MachinePrecision], Or[LessEqual[y, 4.4e+100], N[Not[LessEqual[y, 1.3e+126]], $MachinePrecision]]]], N[(x * N[(y * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999985e-34 or 4.5e-11 < y < 4.4000000000000001e100 or 1.3e126 < y

   1. Initial program 95.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. associate-*l* 85.1%

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

      4. *-commutative 85.1%

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

   6. Simplified 85.1%

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

   if -9.49999999999999985e-34 < y < 4.5e-11 or 4.4000000000000001e100 < y < 1.3e126

   1. Initial program 96.5%

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

      1. distribute-lft-out-- 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. associate-*r* 91.7%

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

   6. Simplified 91.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-34} \lor \neg \left(y \leq 4.5 \cdot 10^{-11}\right) \land \left(y \leq 4.4 \cdot 10^{+100} \lor \neg \left(y \leq 1.3 \cdot 10^{+126}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \]

Alternative 3: 57.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. distribute-lft-out-- 100.0%

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

   2. associate-*r* 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 61.7%

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

   1. *-commutative 61.7%

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

   2. *-commutative 61.7%

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

   3. associate-*l* 61.7%

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

   4. *-commutative 61.7%

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

6. Simplified 61.7%

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

7. Final simplification 61.7%

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

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 2023200 
(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))))