Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.4% → 100.0%

Time: 2.4s

Alternatives: 4

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: 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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 97.6%

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

   1. unpow2 97.6%

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

   2. distribute-lft-out 97.6%

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

   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-rgt-out 100.0%

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

   6. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-29} \lor \neg \left(y \leq 5.7 \cdot 10^{+23}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6e-29) (not (<= y 5.7e+23))) (* 2.0 (* y x)) (* 2.0 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6e-29) || !(y <= 5.7e+23)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = 2.0 * (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 <= 5.7d+23))) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = 2.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6e-29) || !(y <= 5.7e+23)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = 2.0 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6e-29) or not (y <= 5.7e+23):
		tmp = 2.0 * (y * x)
	else:
		tmp = 2.0 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6e-29) || !(y <= 5.7e+23))
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = Float64(2.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6e-29) || ~((y <= 5.7e+23)))
		tmp = 2.0 * (y * x);
	else
		tmp = 2.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6e-29], N[Not[LessEqual[y, 5.7e+23]], $MachinePrecision]], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000005e-29 or 5.7e23 < y

   1. Initial program 94.9%

      \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
   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 82.8%

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

   if -6.0000000000000005e-29 < y < 5.7e23

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 86.8%

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

      1. unpow2 86.8%

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

   6. Simplified 86.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-29} \lor \neg \left(y \leq 5.7 \cdot 10^{+23}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

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

Alternative 4: 59.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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 inf 59.0%

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

   1. unpow2 59.0%

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

6. Simplified 59.0%

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

7. Final simplification 59.0%

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

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 2023199 
(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))))