Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.2% → 100.0%

Time: 4.4s

Alternatives: 5

Speedup: 1.3×

• 28.3% 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: 95.2% 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 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. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

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

   1. metadata-eval 100.0%

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

   2. distribute-lft-neg-in 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. +-commutative 100.0%

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

   6. sub-neg 100.0%

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

   7. associate-*r* 100.0%

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

7. Simplified 100.0%

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

Alternative 2: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.6 \lor \neg \left(y \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.6) (not (<= y 4.6e-7))) (* y (* x -2.0)) (* x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.6) || !(y <= 4.6e-7)) {
		tmp = y * (x * -2.0);
	} 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 <= (-0.6d0)) .or. (.not. (y <= 4.6d-7))) then
        tmp = y * (x * (-2.0d0))
    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 <= -0.6) || !(y <= 4.6e-7)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.6) or not (y <= 4.6e-7):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.6) || !(y <= 4.6e-7))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(x * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.6) || ~((y <= 4.6e-7)))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.6], N[Not[LessEqual[y, 4.6e-7]], $MachinePrecision]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.599999999999999978 or 4.5999999999999999e-7 < y

   1. Initial program 89.8%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. associate-*r* 85.6%

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

      2. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -0.599999999999999978 < y < 4.5999999999999999e-7

   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. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

      1. metadata-eval 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. +-commutative 100.0%

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

      6. sub-neg 100.0%

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

      7. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 89.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.6 \lor \neg \left(y \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.65 \lor \neg \left(y \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.65) (not (<= y 4.9e-5))) (* y (* x -2.0)) (* 2.0 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.65) || !(y <= 4.9e-5)) {
		tmp = y * (x * -2.0);
	} 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 <= (-0.65d0)) .or. (.not. (y <= 4.9d-5))) then
        tmp = y * (x * (-2.0d0))
    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 <= -0.65) || !(y <= 4.9e-5)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = 2.0 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.65) or not (y <= 4.9e-5):
		tmp = y * (x * -2.0)
	else:
		tmp = 2.0 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.65) || !(y <= 4.9e-5))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(2.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.65) || ~((y <= 4.9e-5)))
		tmp = y * (x * -2.0);
	else
		tmp = 2.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.65], N[Not[LessEqual[y, 4.9e-5]], $MachinePrecision]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.650000000000000022 or 4.9e-5 < y

   1. Initial program 89.8%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. associate-*r* 85.6%

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

      2. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -0.650000000000000022 < y < 4.9e-5

   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. Add Preprocessing

   5. Taylor expanded in x around inf 89.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.65 \lor \neg \left(y \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 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. Add Preprocessing
5. Add Preprocessing

Alternative 5: 58.7% 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 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. Add Preprocessing

5. Taylor expanded in x around inf 57.7%

   \[\leadsto 2 \cdot \left(x \cdot \color{blue}{x}\right) \]
6. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* x 2) (- x y)))

  (* 2.0 (- (* x x) (* x y))))