Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.1% → 100.0%

Time: 5.0s

Alternatives: 4

Speedup: 1.3×

• 28.4% 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: 95.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r* 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+36} \lor \neg \left(x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.15e+36) (not (<= x 2e-16)))
   (* x (* x 2.0))
   (* x (* y -2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+36) || !(x <= 2e-16)) {
		tmp = x * (x * 2.0);
	} else {
		tmp = x * (y * -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 ((x <= (-1.15d+36)) .or. (.not. (x <= 2d-16))) then
        tmp = x * (x * 2.0d0)
    else
        tmp = x * (y * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+36) || !(x <= 2e-16)) {
		tmp = x * (x * 2.0);
	} else {
		tmp = x * (y * -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.15e+36) or not (x <= 2e-16):
		tmp = x * (x * 2.0)
	else:
		tmp = x * (y * -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.15e+36) || !(x <= 2e-16))
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = Float64(x * Float64(y * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.15e+36) || ~((x <= 2e-16)))
		tmp = x * (x * 2.0);
	else
		tmp = x * (y * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.15e+36], N[Not[LessEqual[x, 2e-16]], $MachinePrecision]], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14999999999999998e36 or 2e-16 < x

   1. Initial program 91.2%

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

      1. distribute-lft-out-- 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 86.4%

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

      1. unpow2 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

   if -1.14999999999999998e36 < x < 2e-16

   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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. associate-*r* 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-*r* 80.4%

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

   7. Applied egg-rr 80.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+36} \lor \neg \left(x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+35} \lor \neg \left(x \leq 3.7 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8e+35) (not (<= x 3.7e-17))) (* x (* x 2.0)) (* -2.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8e+35) || !(x <= 3.7e-17)) {
		tmp = x * (x * 2.0);
	} else {
		tmp = -2.0 * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-8d+35)) .or. (.not. (x <= 3.7d-17))) then
        tmp = x * (x * 2.0d0)
    else
        tmp = (-2.0d0) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -8e+35) || !(x <= 3.7e-17)) {
		tmp = x * (x * 2.0);
	} else {
		tmp = -2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8e+35) or not (x <= 3.7e-17):
		tmp = x * (x * 2.0)
	else:
		tmp = -2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8e+35) || !(x <= 3.7e-17))
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = Float64(-2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -8e+35) || ~((x <= 3.7e-17)))
		tmp = x * (x * 2.0);
	else
		tmp = -2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8e+35], N[Not[LessEqual[x, 3.7e-17]], $MachinePrecision]], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999997e35 or 3.6999999999999997e-17 < x

   1. Initial program 91.2%

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

      1. distribute-lft-out-- 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 86.4%

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

      1. unpow2 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

   if -7.9999999999999997e35 < x < 3.6999999999999997e-17

   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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 80.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+35} \lor \neg \left(x \leq 3.7 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.8% 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 95.7%

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

   1. distribute-lft-out-- 100.0%

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

   2. associate-*r* 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 52.6%

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

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 2024097 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (* (* x 2.0) (- x y))

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