Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.1% → 100.0%

Time: 4.3s

Alternatives: 4

Speedup: 1.3×

• 28.2% 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

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

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 94.5%

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

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

Alternative 2: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13} \lor \neg \left(y \leq 2.85 \cdot 10^{-74}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.4e-13) (not (<= y 2.85e-74))) (* y (+ x x)) (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e-13) || !(y <= 2.85e-74)) {
		tmp = y * (x + x);
	} else {
		tmp = x * (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 <= (-2.4d-13)) .or. (.not. (y <= 2.85d-74))) then
        tmp = y * (x + x)
    else
        tmp = x * (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e-13) || !(y <= 2.85e-74)) {
		tmp = y * (x + x);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.4e-13) or not (y <= 2.85e-74):
		tmp = y * (x + x)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.4e-13) || !(y <= 2.85e-74))
		tmp = Float64(y * Float64(x + x));
	else
		tmp = Float64(x * Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.4e-13) || ~((y <= 2.85e-74)))
		tmp = y * (x + x);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.4e-13], N[Not[LessEqual[y, 2.85e-74]], $MachinePrecision]], N[(y * N[(x + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3999999999999999e-13 or 2.85000000000000012e-74 < y

   1. Initial program 90.6%

      \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
   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 78.0%

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

      1. associate-*r* 78.0%

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

      2. *-commutative 78.0%

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

      3. associate-*r* 78.0%

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

      4. count-2 78.0%

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

   6. Simplified 78.0%

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

   if -2.3999999999999999e-13 < y < 2.85000000000000012e-74

   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 92.4%

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

      1. unpow2 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*r* 92.4%

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

      4. *-commutative 92.4%

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

      5. count-2 92.4%

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

   6. Simplified 92.4%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13} \lor \neg \left(y \leq 2.85 \cdot 10^{-74}\right):\\ \;\;\;\;y \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 58.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ x x)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (x + x)
end function

Java

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

Python

def code(x, y):
	return x * (x + x)

Julia

function code(x, y)
	return Float64(x * Float64(x + x))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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. Taylor expanded in x around inf 58.6%

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

   1. unpow2 58.6%

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

   2. *-commutative 58.6%

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

   3. associate-*r* 58.6%

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

   4. *-commutative 58.6%

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

   5. count-2 58.6%

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

6. Simplified 58.6%

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

7. Final simplification 58.6%

   \[\leadsto x \cdot \left(x + x\right) \]

Alternative 4: 3.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x y))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x + y

Julia

function code(x, y)
	return Float64(x + y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. associate-*r* 100.0%

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

   2. flip-+ 77.3%

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

   3. associate-*r/ 66.0%

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

   4. *-commutative 66.0%

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

5. Applied egg-rr 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-/l* 75.4%

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

   3. *-commutative 75.4%

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

   4. count-2 75.4%

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

7. Simplified 75.4%

   \[\leadsto \color{blue}{\frac{x \cdot x - y \cdot y}{\frac{x - y}{x + x}}} \]
8. Step-by-step derivation

   1. div-inv 75.4%

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

   2. clear-num 75.8%

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

   3. count-2 75.8%

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

   4. *-un-lft-identity 75.8%

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

   5. times-frac 75.8%

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

   6. metadata-eval 75.8%

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

9. Applied egg-rr 75.8%

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

    1. associate-*r/ 75.8%

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

    2. count-2 75.8%

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

    3. associate-*r/ 66.0%

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

    4. difference-of-squares 74.2%

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

    5. associate-*r* 80.6%

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

    6. associate-/l* 99.9%

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

    7. +-commutative 99.9%

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

    8. associate-/r* 99.7%

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

    9. *-inverses 99.7%

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

    10. count-2 99.7%

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

    11. associate-/r* 99.7%

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

    12. metadata-eval 99.7%

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

11. Simplified 99.7%

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

13. Applied egg-rr 12.2%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + y}\right)}^{2}}{x + x} \cdot \left(\sqrt[3]{x + y} \cdot \left(x + x\right)\right)} \]
14. Step-by-step derivation

    1. associate-*r* 3.9%

       \[\leadsto \color{blue}{\left(\frac{{\left(\sqrt[3]{x + y}\right)}^{2}}{x + x} \cdot \sqrt[3]{x + y}\right) \cdot \left(x + x\right)} \]

    2. associate-*l/ 3.9%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + y}\right)}^{2} \cdot \sqrt[3]{x + y}}{x + x}} \cdot \left(x + x\right) \]

    3. unpow2 3.9%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right)} \cdot \sqrt[3]{x + y}}{x + x} \cdot \left(x + x\right) \]

    4. rem-3cbrt-lft 3.9%

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

    5. associate-/r/ 3.9%

       \[\leadsto \color{blue}{\frac{x + y}{\frac{x + x}{x + x}}} \]

    6. *-inverses 3.9%

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

    7. /-rgt-identity 3.9%

       \[\leadsto \color{blue}{x + y} \]

15. Simplified 3.9%

    \[\leadsto \color{blue}{x + y} \]

16. Final simplification 3.9%

    \[\leadsto x + y \]

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 2023192 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (* x 2.0) (+ x y))

  (* 2.0 (+ (* x x) (* x y))))