Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.1% → 100.0%

Time: 2.6s

Alternatives: 3

Speedup: 1.3×

• 28.2% 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: 95.1% 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 94.5%

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

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

Alternative 2: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13} \lor \neg \left(y \leq 2.85 \cdot 10^{-74}\right) \land \left(y \leq 3.7 \cdot 10^{-11} \lor \neg \left(y \leq 5.6 \cdot 10^{+76}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.4e-13)
         (and (not (<= y 2.85e-74)) (or (<= y 3.7e-11) (not (<= y 5.6e+76)))))
   (* y (* x -2.0))
   (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e-13) || (!(y <= 2.85e-74) && ((y <= 3.7e-11) || !(y <= 5.6e+76)))) {
		tmp = y * (x * -2.0);
	} 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 <= (-2.4d-13)) .or. (.not. (y <= 2.85d-74)) .and. (y <= 3.7d-11) .or. (.not. (y <= 5.6d+76))) then
        tmp = y * (x * (-2.0d0))
    else
        tmp = x * (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e-13) || (!(y <= 2.85e-74) && ((y <= 3.7e-11) || !(y <= 5.6e+76)))) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.4e-13) or (not (y <= 2.85e-74) and ((y <= 3.7e-11) or not (y <= 5.6e+76))):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.4e-13) || (!(y <= 2.85e-74) && ((y <= 3.7e-11) || !(y <= 5.6e+76))))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(x * Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.4e-13) || (~((y <= 2.85e-74)) && ((y <= 3.7e-11) || ~((y <= 5.6e+76)))))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.4e-13], And[N[Not[LessEqual[y, 2.85e-74]], $MachinePrecision], Or[LessEqual[y, 3.7e-11], N[Not[LessEqual[y, 5.6e+76]], $MachinePrecision]]]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3999999999999999e-13 or 2.85000000000000012e-74 < y < 3.7000000000000001e-11 or 5.5999999999999997e76 < y

   1. Initial program 90.7%

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

   2. Taylor expanded in x around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*r* 83.2%

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

   4. Simplified 83.2%

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

   if -2.3999999999999999e-13 < y < 2.85000000000000012e-74 or 3.7000000000000001e-11 < y < 5.5999999999999997e76

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 91.4%

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

      1. unpow2 91.4%

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

      2. count-2 91.4%

         \[\leadsto \color{blue}{x \cdot x + x \cdot x} \]

      3. distribute-lft-in 91.4%

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

   4. Simplified 91.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13} \lor \neg \left(y \leq 2.85 \cdot 10^{-74}\right) \land \left(y \leq 3.7 \cdot 10^{-11} \lor \neg \left(y \leq 5.6 \cdot 10^{+76}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 58.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

2. Taylor expanded in x around inf 57.9%

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

   1. unpow2 57.9%

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

   2. count-2 57.9%

      \[\leadsto \color{blue}{x \cdot x + x \cdot x} \]

   3. distribute-lft-in 57.9%

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

4. Simplified 57.9%

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

5. Final simplification 57.9%

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

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 2023192 
(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))))