Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.5% → 100.0%

Time: 4.5s

Alternatives: 3

Speedup: 1.3×

• 27.8% 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: 95.5% 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 90.2%

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

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

Alternative 2: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+252}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e+252) (* -2.0 (* x y)) (* y (* 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+252) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = y * (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 (x <= 4d+252) then
        tmp = (-2.0d0) * (x * y)
    else
        tmp = y * (2.0d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4e+252) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = y * (2.0 * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e+252:
		tmp = -2.0 * (x * y)
	else:
		tmp = y * (2.0 * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+252)
		tmp = Float64(-2.0 * Float64(x * y));
	else
		tmp = Float64(y * Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4e+252)
		tmp = -2.0 * (x * y);
	else
		tmp = y * (2.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+252], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000004e252

   1. Initial program 92.6%

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

   5. Taylor expanded in x around 0 58.7%

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 58.7%

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

      2. *-commutative 58.7%

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

      3. distribute-rgt-neg-in 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in y around 0 58.7%

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

   if 4.0000000000000004e252 < x

   1. Initial program 41.7%

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

      1. distribute-lft-out-- 100.0%

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

   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 17.5%

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 17.5%

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

      2. *-commutative 17.5%

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

      3. distribute-rgt-neg-in 17.5%

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

   7. Simplified 17.5%

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

      1. add-log-exp 33.3%

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

      2. *-commutative 33.3%

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

      3. exp-lft-sqr 33.3%

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

      4. log-prod 33.3%

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

      5. add-log-exp 33.3%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 16.7%

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

      10. add-sqr-sqrt 16.7%

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

      11. add-log-exp 0.4%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 66.7%

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

      14. sqr-neg 66.7%

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

      15. sqrt-unprod 58.9%

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

      16. add-sqr-sqrt 58.9%

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

   9. Applied egg-rr 58.9%

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

       1. distribute-lft-out 58.9%

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

       2. count-2 58.9%

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

       3. *-commutative 58.9%

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

   11. Simplified 58.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+252}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% 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 90.2%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 0 56.8%

   \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 56.8%

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

   2. *-commutative 56.8%

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

   3. distribute-rgt-neg-in 56.8%

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

7. Simplified 56.8%

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

8. Taylor expanded in y around 0 56.8%

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

9. Final simplification 56.8%

   \[\leadsto -2 \cdot \left(x \cdot y\right) \]
10. 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 2024071 
(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))))