Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.3% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 1.3×

• 28.0% 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: 95.3% 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

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

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]

Derivation

1. Initial program 96.9%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 96.9%

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

   1. metadata-eval 96.9%

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

   2. distribute-lft-neg-in 96.9%

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

   3. associate-*l* 96.5%

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

   4. distribute-rgt-neg-out 96.5%

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

   5. unpow2 96.5%

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

   6. associate-*l* 96.5%

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

   7. distribute-lft-out 100.0%

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

   8. *-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-37} \lor \neg \left(y \leq 3.1 \cdot 10^{-21}\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e-37) (not (<= y 3.1e-21))) (* -2.0 (* x y)) (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e-37) || !(y <= 3.1e-21)) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7d-37)) .or. (.not. (y <= 3.1d-21))) then
        tmp = (-2.0d0) * (x * y)
    else
        tmp = x * (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e-37) || !(y <= 3.1e-21)) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e-37) or not (y <= 3.1e-21):
		tmp = -2.0 * (x * y)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e-37) || !(y <= 3.1e-21))
		tmp = Float64(-2.0 * Float64(x * y));
	else
		tmp = Float64(x * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e-37) || ~((y <= 3.1e-21)))
		tmp = -2.0 * (x * y);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e-37], N[Not[LessEqual[y, 3.1e-21]], $MachinePrecision]], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000003e-37 or 3.0999999999999998e-21 < y

   1. Initial program 94.2%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 82.6%

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

   if -7.0000000000000003e-37 < y < 3.0999999999999998e-21

   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 90.4%

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

      1. unpow2 90.4%

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

      2. *-commutative 90.4%

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

      3. associate-*r* 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-37} \lor \neg \left(y \leq 3.1 \cdot 10^{-21}\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \]

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 96.9%

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

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

Alternative 4: 56.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 58.9%

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

5. Final simplification 58.9%

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

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 2023275 
(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))))