Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 5.9s

Alternatives: 5

Speedup: 1.3×

• 27.9% 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

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

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 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 2: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-68} \lor \neg \left(x \leq 1.25 \cdot 10^{-70}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.7e-68) (not (<= x 1.25e-70)))
   (* 2.0 (* x x))
   (* y (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.7e-68) || !(x <= 1.25e-70)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (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 ((x <= (-2.7d-68)) .or. (.not. (x <= 1.25d-70))) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = y * (x * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.7e-68) || !(x <= 1.25e-70)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (x * -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.7e-68) or not (x <= 1.25e-70):
		tmp = 2.0 * (x * x)
	else:
		tmp = y * (x * -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.7e-68) || !(x <= 1.25e-70))
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(y * Float64(x * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.7e-68) || ~((x <= 1.25e-70)))
		tmp = 2.0 * (x * x);
	else
		tmp = y * (x * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.7e-68], N[Not[LessEqual[x, 1.25e-70]], $MachinePrecision]], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002e-68 or 1.25e-70 < x

   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. Add Preprocessing

   5. Taylor expanded in x around inf 81.4%

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

   if -2.7000000000000002e-68 < x < 1.25e-70

   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 99.1%

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

      1. *-commutative 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. *-commutative 99.1%

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

      6. distribute-rgt-out 99.1%

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

      7. +-commutative 99.1%

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

      8. sub-neg 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around 0 97.1%

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

      1. associate-*r* 97.1%

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

      2. *-commutative 97.1%

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

      3. *-commutative 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-68} \lor \neg \left(x \leq 1.25 \cdot 10^{-70}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-67} \lor \neg \left(x \leq 5.5 \cdot 10^{-71}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.55e-67) (not (<= x 5.5e-71)))
   (* 2.0 (* x x))
   (* -2.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.55e-67) || !(x <= 5.5e-71)) {
		tmp = 2.0 * (x * x);
	} 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 <= (-2.55d-67)) .or. (.not. (x <= 5.5d-71))) then
        tmp = 2.0d0 * (x * x)
    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 <= -2.55e-67) || !(x <= 5.5e-71)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = -2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.55e-67) or not (x <= 5.5e-71):
		tmp = 2.0 * (x * x)
	else:
		tmp = -2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.55e-67) || !(x <= 5.5e-71))
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(-2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.55e-67) || ~((x <= 5.5e-71)))
		tmp = 2.0 * (x * x);
	else
		tmp = -2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.55e-67], N[Not[LessEqual[x, 5.5e-71]], $MachinePrecision]], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999991e-67 or 5.4999999999999997e-71 < x

   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. Add Preprocessing

   5. Taylor expanded in x around inf 81.4%

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

   if -2.54999999999999991e-67 < x < 5.4999999999999997e-71

   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 97.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-67} \lor \neg \left(x \leq 5.5 \cdot 10^{-71}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \]
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 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 56.4%

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

Alternative 5: 10.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 96.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

   4. distribute-lft-neg-out 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-out 99.6%

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

   7. +-commutative 99.6%

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

   8. sub-neg 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around inf 62.5%

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

   1. add-sqr-sqrt 33.0%

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

   2. sqrt-unprod 39.4%

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

   3. pow1/2 39.4%

      \[\leadsto x \cdot \color{blue}{{\left(\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)\right)}^{0.5}} \]

10. Applied egg-rr 13.5%

    \[\leadsto x \cdot \color{blue}{{\log \left({1}^{x}\right)}^{0.5}} \]
11. Step-by-step derivation

    1. pow-base-1 13.5%

       \[\leadsto x \cdot {\log \color{blue}{1}}^{0.5} \]

    2. metadata-eval 13.5%

       \[\leadsto x \cdot {\color{blue}{0}}^{0.5} \]

    3. metadata-eval 13.5%

       \[\leadsto x \cdot \color{blue}{0} \]

12. Simplified 13.5%

    \[\leadsto x \cdot \color{blue}{0} \]

13. Taylor expanded in x around 0 13.5%

    \[\leadsto \color{blue}{0} \]
14. 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 2024170 
(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))))