Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.4% → 100.0%

Time: 3.2s

Alternatives: 3

Speedup: 1.3×

• 27.6% 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: 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

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

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 94.5%

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

   1. *-commutative 94.5%

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

   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: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1260000000000 \lor \neg \left(y \leq 4.5 \cdot 10^{+50}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1260000000000.0) (not (<= y 4.5e+50)))
   (* y (+ x x))
   (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1260000000000.0) || !(y <= 4.5e+50)) {
		tmp = y * (x + x);
	} 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 <= (-1260000000000.0d0)) .or. (.not. (y <= 4.5d+50))) then
        tmp = y * (x + x)
    else
        tmp = x * (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1260000000000.0) || !(y <= 4.5e+50)) {
		tmp = y * (x + x);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1260000000000.0) or not (y <= 4.5e+50):
		tmp = y * (x + x)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1260000000000.0) || !(y <= 4.5e+50))
		tmp = Float64(y * Float64(x + x));
	else
		tmp = Float64(x * Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1260000000000.0) || ~((y <= 4.5e+50)))
		tmp = y * (x + x);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1260000000000.0], N[Not[LessEqual[y, 4.5e+50]], $MachinePrecision]], N[(y * N[(x + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.26e12 or 4.50000000000000014e50 < y

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*r* 86.9%

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

      3. *-commutative 86.9%

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

      4. count-2 86.9%

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

   4. Simplified 86.9%

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

   if -1.26e12 < y < 4.50000000000000014e50

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.4%

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

      1. unpow2 84.4%

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

      2. *-commutative 84.4%

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

      3. associate-*r* 84.4%

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

      4. *-commutative 84.4%

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

      5. count-2 84.4%

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

   4. Simplified 84.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1260000000000 \lor \neg \left(y \leq 4.5 \cdot 10^{+50}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 58.7% 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 59.3%

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

   1. unpow2 59.3%

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

   2. *-commutative 59.3%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. count-2 59.3%

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

4. Simplified 59.3%

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

5. Final simplification 59.3%

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

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 2023279 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

  (* 2.0 (+ (* x x) (* x y))))