Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.5% → 100.0%

Time: 5.1s

Alternatives: 4

Speedup: 1.3×

• 27.2% 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: 95.5% 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 95.7%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 95.7%

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

   1. unpow2 95.7%

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

   2. distribute-lft-out 95.7%

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

   3. distribute-rgt-in 100.0%

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

   4. count-2 100.0%

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

   5. distribute-lft-out 100.0%

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

   6. count-2 100.0%

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

   7. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-13} \lor \neg \left(y \leq 4.3 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e-13) (not (<= y 4.3e+57))) (* y (+ x x)) (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e-13) || !(y <= 4.3e+57)) {
		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 <= (-2.1d-13)) .or. (.not. (y <= 4.3d+57))) 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 <= -2.1e-13) || !(y <= 4.3e+57)) {
		tmp = y * (x + x);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e-13) or not (y <= 4.3e+57):
		tmp = y * (x + x)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e-13) || !(y <= 4.3e+57))
		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 <= -2.1e-13) || ~((y <= 4.3e+57)))
		tmp = y * (x + x);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e-13], N[Not[LessEqual[y, 4.3e+57]], $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 < -2.09999999999999989e-13 or 4.30000000000000033e57 < y

   1. Initial program 90.6%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 83.9%

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

      1. associate-*r* 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*r* 83.9%

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

      4. count-2 83.9%

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

   6. Simplified 83.9%

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

   if -2.09999999999999989e-13 < y < 4.30000000000000033e57

   1. Initial program 99.3%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 86.5%

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

      1. unpow2 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. *-commutative 86.5%

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

      5. count-2 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-13} \lor \neg \left(y \leq 4.3 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 2.0 * (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 + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 58.8% 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 95.7%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 61.9%

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

   1. unpow2 61.9%

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

   2. *-commutative 61.9%

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

   3. associate-*r* 61.9%

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

   4. *-commutative 61.9%

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

   5. count-2 61.9%

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

6. Simplified 61.9%

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

7. Final simplification 61.9%

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

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 2023278 
(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))))