Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.0% → 100.0%

Time: 4.2s

Alternatives: 3

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: 95.0% 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 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 2: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+107} \lor \neg \left(y \leq -5.8 \cdot 10^{+54}\right) \land \left(y \leq -1.9 \cdot 10^{-74} \lor \neg \left(y \leq 7 \cdot 10^{+29}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e+107)
         (and (not (<= y -5.8e+54)) (or (<= y -1.9e-74) (not (<= y 7e+29)))))
   (* y (* x -2.0))
   (* x (* 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+107) || (!(y <= -5.8e+54) && ((y <= -1.9e-74) || !(y <= 7e+29)))) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (2.0 * 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 <= (-1.9d+107)) .or. (.not. (y <= (-5.8d+54))) .and. (y <= (-1.9d-74)) .or. (.not. (y <= 7d+29))) then
        tmp = y * (x * (-2.0d0))
    else
        tmp = x * (2.0d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+107) || (!(y <= -5.8e+54) && ((y <= -1.9e-74) || !(y <= 7e+29)))) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (2.0 * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e+107) or (not (y <= -5.8e+54) and ((y <= -1.9e-74) or not (y <= 7e+29))):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (2.0 * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e+107) || (!(y <= -5.8e+54) && ((y <= -1.9e-74) || !(y <= 7e+29))))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(x * Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e+107) || (~((y <= -5.8e+54)) && ((y <= -1.9e-74) || ~((y <= 7e+29)))))
		tmp = y * (x * -2.0);
	else
		tmp = x * (2.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e+107], And[N[Not[LessEqual[y, -5.8e+54]], $MachinePrecision], Or[LessEqual[y, -1.9e-74], N[Not[LessEqual[y, 7e+29]], $MachinePrecision]]]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e107 or -5.7999999999999997e54 < y < -1.8999999999999998e-74 or 6.99999999999999958e29 < y

   1. Initial program 92.9%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 86.5%

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

      1. associate-*r* 86.5%

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

   6. Simplified 86.5%

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

   if -1.8999999999999999e107 < y < -5.7999999999999997e54 or -1.8999999999999998e-74 < y < 6.99999999999999958e29

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 93.7%

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

      2. pow2 93.7%

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

   5. Applied egg-rr 93.7%

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

      1. unpow2 93.7%

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

      2. add-sqr-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. remove-double-div 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{2 \cdot \left(x - y\right)}}} \cdot x \]

      6. associate-/l/ 99.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{x - y}}{2}}} \cdot x \]

      7. associate-/r/ 99.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{1}{x - y}}{2}}{x}}} \]

      8. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{x - y}}{2}}} \]

      9. div-inv 99.7%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{x - y} \cdot \frac{1}{2}}} \]

      10. metadata-eval 99.7%

          \[\leadsto \frac{x}{\frac{1}{x - y} \cdot \color{blue}{0.5}} \]

      11. associate-*l/ 99.7%

          \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 0.5}{x - y}}} \]

      12. metadata-eval 99.7%

          \[\leadsto \frac{x}{\frac{\color{blue}{0.5}}{x - y}} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5}{x - y}}} \]

   8. Taylor expanded in x around inf 90.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{0.5}{x}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 90.6%

         \[\leadsto \color{blue}{\frac{x}{0.5} \cdot x} \]

      2. div-inv 90.6%

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{0.5}\right)} \cdot x \]

      3. metadata-eval 90.6%

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

   10. Applied egg-rr 90.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+107} \lor \neg \left(y \leq -5.8 \cdot 10^{+54}\right) \land \left(y \leq -1.9 \cdot 10^{-74} \lor \neg \left(y \leq 7 \cdot 10^{+29}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \end{array} \]

Alternative 3: 57.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 61.0%

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

   1. associate-*r* 61.0%

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

6. Simplified 61.0%

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

7. Final simplification 61.0%

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

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 2023308 
(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))))