Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 94.7% → 100.0%

Time: 4.0s

Alternatives: 5

Speedup: 1.3×

• 28.5% 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

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

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 96.4%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-29} \lor \neg \left(y \leq 6.8 \cdot 10^{+21}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6e-29) (not (<= y 6.8e+21))) (* y (* x -2.0)) (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6e-29) || !(y <= 6.8e+21)) {
		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 <= (-6d-29)) .or. (.not. (y <= 6.8d+21))) 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 <= -6e-29) || !(y <= 6.8e+21)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6e-29) or not (y <= 6.8e+21):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6e-29) || !(y <= 6.8e+21))
		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 <= -6e-29) || ~((y <= 6.8e+21)))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6e-29], N[Not[LessEqual[y, 6.8e+21]], $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 < -6.0000000000000005e-29 or 6.8e21 < y

   1. Initial program 92.4%

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

   2. Taylor expanded in x around 0 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.5%

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

   4. Simplified 84.5%

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

   if -6.0000000000000005e-29 < y < 6.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 86.9%

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

      1. unpow2 86.9%

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

      2. count-2 86.9%

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

      3. distribute-lft-in 86.9%

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

   4. Simplified 86.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-29} \lor \neg \left(y \leq 6.8 \cdot 10^{+21}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 59.3% 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 96.4%

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

2. Taylor expanded in x around inf 57.2%

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

   1. unpow2 57.2%

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

   2. count-2 57.2%

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

   3. distribute-lft-in 57.2%

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

4. Simplified 57.2%

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

5. Final simplification 57.2%

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

Alternative 4: 15.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. flip-- 85.6%

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

   4. associate-*r/ 73.5%

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

   5. add-log-exp 36.1%

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

   6. *-commutative 36.1%

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

   7. exp-lft-sqr 36.1%

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

   8. log-prod 36.1%

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

   9. add-log-exp 40.5%

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

   10. add-log-exp 73.5%

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

3. Applied egg-rr 73.5%

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

   1. associate-/l* 85.4%

      \[\leadsto \color{blue}{\frac{x + x}{\frac{x + y}{x \cdot x - y \cdot y}}} \]

   2. difference-of-squares 89.3%

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

   3. associate-/r* 99.7%

      \[\leadsto \frac{x + x}{\color{blue}{\frac{\frac{x + y}{x + y}}{x - y}}} \]

   4. *-inverses 99.7%

      \[\leadsto \frac{x + x}{\frac{\color{blue}{1}}{x - y}} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{x + x}{\frac{1}{x - y}}} \]

7. Applied egg-rr 15.9%

   \[\leadsto \color{blue}{y \cdot y} \]

8. Final simplification 15.9%

   \[\leadsto y \cdot y \]

Alternative 5: 11.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 96.4%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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. flip-- 85.6%

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

   4. associate-*r/ 73.5%

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

   5. add-log-exp 36.1%

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

   6. *-commutative 36.1%

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

   7. exp-lft-sqr 36.1%

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

   8. log-prod 36.1%

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

   9. add-log-exp 40.5%

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

   10. add-log-exp 73.5%

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

3. Applied egg-rr 73.5%

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

   1. associate-/l* 85.4%

      \[\leadsto \color{blue}{\frac{x + x}{\frac{x + y}{x \cdot x - y \cdot y}}} \]

   2. difference-of-squares 89.3%

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

   3. associate-/r* 99.7%

      \[\leadsto \frac{x + x}{\color{blue}{\frac{\frac{x + y}{x + y}}{x - y}}} \]

   4. *-inverses 99.7%

      \[\leadsto \frac{x + x}{\frac{\color{blue}{1}}{x - y}} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{x + x}{\frac{1}{x - y}}} \]

7. Applied egg-rr 14.8%

   \[\leadsto \color{blue}{y - y} \]
8. Step-by-step derivation

   1. +-inverses 14.8%

      \[\leadsto \color{blue}{0} \]

9. Simplified 14.8%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 14.8%

    \[\leadsto 0 \]

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 2023199 
(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))))